Factors determining the precision of the correlated firing generated by a monosynaptic connection in the cat visual pathway
1 Departamento de Lenguajes y Ciencias de la Computación, Universidad de Málaga, Málaga, Spain
2 Department of Psychology, University of Connecticut, Storrs, CT, USA
3 SUNY State College of Optometry, New York, NY, USA
Abstract
Across the visual pathway, strong monosynaptic connections generate a precise correlated firing between presynaptic and postsynaptic neurons. The precision of this correlated firing is not the same within thalamus and visual cortex. While retinogeniculate connections generate a very narrow peak in the correlogram (peak width < 1 ms), the peaks generated by geniculocortical and corticocortical connections have usually a time course of several milliseconds. Several factors could explain these differences in timing precision such as the amplitude of the monosynaptic EPSP (excitatory postsynaptic potential), its time course or the contribution of polysynaptic inputs. While it is difficult to isolate the contribution of each factor in physiological experiments, a first approximation can be done in modelling studies. Here, we simulated two monosynaptically connected neurons to measure changes in their correlated firing as we independently modified different parameters of the connection. Our results suggest that the precision of the correlated firing generated by strong monosynaptic connections is mostly determined by the EPSP time course of the connection and much less by other factors. In addition, we show that a polysynaptic pathway is unlikely to emulate the correlated firing generated by a monosynaptic connection unless it generates EPSPs with very small latency jitter.
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Introduction
Neuronal response properties undergo several important transformations from the retina to the primary visual cortex. Receptive fields become more elaborated (Hubel & Wiesel, 1962), average firing rates are reduced and visual responses become more variable (Kara et al. 2000). Studies of cross-correlation analysis suggest an additional transformation – the correlated firing between presynaptic and postsynaptic neurons becomes less precise. While retinogeniculate connections generate very narrow correlograms with less than 1 ms width at half-amplitude (Levick et al. 1972; Mastronarde, 1987; Usrey et al. 1999), the connections from the lateral geniculate nucleus (LGN) to cortical layer 4 and from layer 4 to layers 2 and 3 generate much wider correlograms (Toyama et al. 1981; Tanaka, 1983; Reid & Alonso, 1995; Alonso & Martinez, 1998; Alonso et al. 2001).
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It is unclear what factors are responsible for these differences in correlated firing precision. Retinogeniculate connections differ from geniculocortical and corticocortical connections in a large variety of anatomical and physiological factors that could be important in determining precision. While a retinal afferent makes more than a hundred synapses with a single geniculate cell (Hamos et al. 1987), a geniculate or a cortical afferent makes usually less than 10 synapses per neuronal target (Freund et al. 1985; Feldmeyer et al. 2002). Retinogeniculate EPSPs (excitatory postsynaptic potentials) are also larger, faster, shorter and have less jitter in their latency than corticocortical EPSPs (e.g. Eysel, 1976; Bloomfield & Sherman, 1988; Stratford et al. 1996). Moreover, while one to three retinal afferents converge into a single geniculate neuron, a layer 4 neuron may receive convergent input from around 30 geniculate afferents (Cleland et al. 1971a; Chen & Regehr, 2000; Alonso et al. 2001). Cortical neurons are also heavily interconnected while geniculate cells barely make excitatory connections with each other (Friedlander et al. 1979).
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Previous studies in the motor and somatosensory systems did not reach an agreement on whether the width of a monosynaptic peak in a correlogram is determined by the EPSP rise time (Knox, 1974; Fetz & Gustafsson, 1983), the EPSP duration (Moore et al. 1970; Lindsey & Gerstein, 1979; Surmeier & Weinberg, 1985) or a linear combination of both (Kirkwood & Sears, 1982; Gustafsson & McCrea, 1984; Cope et al. 1987). Moreover, although the contribution of other parameters such as the synaptic noise and threshold were thought to have an important contribution, they were not explored in detail (see Kirkwood, 1979 for review). Here, we have systematically explored the contribution of a large number of synaptic parameters that could determine the width of a monosynaptic correlogram in the visual pathway. In contrast to previous studies (Surmeier & Weinberg, 1985), we were able to modify different parameters independently (e.g. EPSP rise time and duration). Moreover, our simulations used parameter values consistent with synaptic physiology measurements (Eysel, 1976; Bloomfield & Sherman, 1988; Stratford et al. 1996) and generated realistic correlograms that resemble very closely those measured at three different stages of the visual pathway (retina–LGN (Levick et al. 1972; Mastronarde, 1987; Usrey et al. 1999); LGN–cortex (Tanaka, 1983; Reid & Alonso, 1995; Alonso et al. 2001); cortical layer 4–superficial layers of cortex (Alonso & Martinez, 1998)).
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The precise correlated firing generated by presynaptic and postsynaptic neurons plays a major role in the development and plastic remodelling of the visual pathway (Katz & Shatz, 1996) since the relative timing of discharges is a crucial parameter (Bienenstock et al. 1982; Bell et al. 1997; Markram et al. 1997b; Bi & Poo, 1998; Debanne et al. 1998; Song et al. 2000; Sjostrom et al. 2001; Froemke & Dan, 2002; Fu et al. 2002). Therefore, it is important to identify the factors that are responsible for the differences in correlated-firing precision. Our results suggest that the most important factors are the EPSP time course (rise time, duration and latency jitter). EPSPs with large values of duration and latency jitter generate broad correlogram peaks and those with small values generate narrow correlogram peaks. This finding resonates quite well with recent results in the hippocampus, which indicate that spike-timing precision is strongly influenced by the shape of the compound EPSP (Fricker & Miles, 2000; Pouille & Scanziani, 2001; Axmacher & Miles, 2004; see also Poliakov et al. 1997). For example, CA1 inhibitory neurons, which generate short-duration EPSPs, initiate most of their spikes at the rising phase of the EPSP. In contrast, CA1 pyramidal neurons, which generate prolonged EPSPs (due to the activation of a Na+ current) initiate their spikes both at the rising and decay phase of the EPSP (Fricker & Miles, 2000). Spikes are also most frequently generated at the rising phase of the EPSP when the compound EPSP is shorten by strong sequences of outward–inward currents (Axmacher & Miles, 2004) or strong IPSPs (Pouille & Scanziani, 2001; Wehr & Zador, 2003). Based on these experimental data and our results, we propose that the EPSP time course increases from thalamus to visual cortex, reducing the precision of the monosynaptic correlated firing while making temporal summation more effective.
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Methods
Throughout this paper we use two simulated neurons that are monosynaptically connected: N1 is the presynaptic neuron and N2 is the postsynaptic neuron. In addition to the input from N1, synaptic noise is injected in the membrane potential of N2 to simulate the contribution of other inputs that are not N1. Synaptic noise is also injected in N1 to generate presynaptic spikes. The monosynaptic connection generates a precise correlated firing that we measure with cross-correlation analysis. The precision of the correlated firing is estimated from the width of the monosynaptic peak in the correlogram. We measure how the peak width is modified as we change different parameters of the monosynaptic connection. In addition, we measure changes in the rise time of the monosynaptic peak, which are summarized in Table 1.
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The neuron model
Compartmental models take into account the geometrical and physiological characteristics of the neuron, like axons, dendritic spines, ionic channels or soma. On the other hand, point models characterize the neuron's electrical behaviour, and consider it as a single point in space (Hodgkin & Huxley, 1952; FitzHugh, 1961; Nagumo et al. 1962; Morris & Lecar, 1981). These simplified models can introduce the main features of neuronal physiology with a relatively low computational cost. Integrate-and-fire (IAF) neurons are a particular case of simplified models that derive from the pioneering work of Louis Lapicque (Lapicque, 1907, 1926). Different versions of this model have been proposed in various studies (Stein, 1967a,b; Knight, 1972; Jack et al. 1983; Tuckwell, 1988). The traditional form of an IAF model is a first-order differential equation (eqn (1)) whose dynamics are characterized by a subthreshold integration domain (where the neuron integrates the inputs and behaves like a RC circuit) and a threshold voltage for the generation of action potentials.
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In eqn (1), Rm and Cm are the neuronal membrane resistance and capacitance, respectively, Vm is the membrane potential, Vrest is the resting potential and I(t) is the input current. In eqn (2), the input current I(t) is determined by the neuronal membrane potential Vm (given a reversal potential of Esyn) and the synaptic conductance gsyn. The term j represents the synaptic efficacy of each connection and noise is the background neuronal activity. The synaptic conductance gsyn (eqn (3)) was modelled with two alpha functions. This approach allowed us to independently control the rise and decay times of the conductance by adjusting two time constants: trise and tdecay. The conductance function starts at the time of the presynaptic spike j (when ), which is determined by the synaptic latency and latency jitter between the presynaptic and the postsynaptic neuron. Inhibitory synapses are included in the model by assigning negative values to j. The synaptic noise is modelled by creating random negative and positive deflections in the amplitude of Vm (the distribution of the noise-event amplitudes can be gaussian or uniform). Ideally, we would reproduce the noise of each neuron by using a sum of realistic EPSPs with different amplitudes and time courses. Unfortunately, we do not know enough about the characteristics of the multiple unitary EPSPs that make the background activity of the different neurons within the visual pathway. Consequently, we decided to use a more general form of random noise, which allows us to investigate the effect of modifying the amplitude distribution and time course of the noise events (Figs 6 and 7).
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Correlograms (N1 N2) obtained with different gaussian distributions of synaptic-event amplitudes, called here synaptic noise (average amplitude is constant; S.D. varies). The cartoon on the top-left illustrates the broadening of the distributions of noise amplitude. Increasing the S.D. of the noise made monosynaptic peaks slightly broader and in amplitude. Total number of N2 spikes: 10 000 for all three correlograms. On the left, examples of the EPSPs + synaptic noise used (shown as spike trigger averages; total time shown, 20 ms; EPSP amplitude, 3.6 mV). Symbols as in Fig. 3. Firing rates: N1, 75 spikes s–1; N2, 115–225 spikes s–1.
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Correlograms (N1 N2) obtained with different frequency spectrums of synaptic noise. The synaptic noise has been filtered by convolving it with exponential functions with different time constants (average noise and S.D. are constant; time constant varies). Modifying the frequency spectrum of the synaptic noise had a small influence on the width of the monosynaptic peak. Notice that, as the noise is smoothed, the correlograms start showing multiple peaks. These non-physiological correlogram shapes are also found when the noise is totally eliminated (see Fig. 12). On the left, examples of EPSPs + synaptic noise (shown as spike trigger averages; total time shown, 20 ms; EPSP amplitude, 3.6 mV). Symbols as in Fig. 3. Firing rates: N1, 100 spikes s–1; N2, 140 spikes s–1.
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With these three equations we define a new IAF model, based on synaptic currents, that includes some of the main physiological parameters that influence the time precision of action potentials and the shape of the monosynaptic crosscorrelogram. The model allowed us to manipulate 10 parameters independently for each neuron: firing rate, synaptic noise, EPSP rise time, EPSP duration at half-amplitude, EPSP amplitude, latency jitter, conduction delay, resting potential, threshold and refractory period. In most simulations only one parameter was modified while the others were kept constant at their default values. The default values were as follows: N1 firing rate: 25–100 spikes s–1; N1 resting potential: –70 mV; N1 threshold: –40 mV; N1 uniform noise: 2.1–3.6 nA; N2 resting potential: –70 mV; N2 threshold: –40 mV; N2 gaussian noise: μ = 2.6 nA, s = 0.5 nA; N2 EPSP rise time: 2.9 ms; N2 EPSP duration at half-amplitude: 8.7 ms; N2 EPSP amplitude (measured at the resting potential): 2.35 mV. The EPSP amplitude depends on the membrane potential. It is 0 for a membrane potential of 0 mV and maximum for a membrane potential of –80 mV. The values of EPSP rise time, duration and amplitude were all measured at the resting potential. The EPSP rise time was measured from the baseline to the peak of the EPSP. The EPSP duration was measured as the width of the EPSP at half-amplitude; N1 N2 weight (synaptic efficacy modelled as maximum conductance in eqn (2)): 8.75 nS for simulations using gaussian noise and 6.25 nS for simulations using uniform noise. The connection strength was defined by the weight. For a weight of 1.25 nS, the EPSP amplitude was 0.77 mV (measured at resting potential); N1 N2 synaptic delay: 1 ms; N1 N2 latency jitter: 0.5 ms.
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In some experiments we used three different default values to explore how pair combinations of parameters influence the shape of the monosynaptic peak in the correlogram. In these experiments we used the following default values: EPSP rise time: 0.5, 1.7 and 3.1 ms; EPSP duration: 2.4, 14.4 and 30.3 ms; threshold: –38.5, –37 mV (–64.75, –64.5 mV when the test parameter is EPSP rise time); synaptic noise: 0, 0.5 and 1 nA; EPSP/noise: 0.85, 2.47 and 5.26; latency jitter: 0, 0.8 and 1.5 ms.
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The parameter values explored in our simulations are inspired by physiological measurements of different connections across the early visual pathway. For example, according to Bloomfield & Sherman (1988), retinogeniculate EPSPs in the cat have mean amplitudes of 1.87 mV for X cells and 2.6 mV for Y cells, rise times of 0.61 ms for X cells and 0.43 ms for Y cells, total durations of 1.91 ms for X cells and 3.21 for Y cells and latency jitters of 0.35 ms for X cells and 0.17 for Y cells (see also Eysel, 1976). Stratford et al. (1996) measured cortical EPSPs in the cat that were thought to originate in thalamus and cortex. They measured the following values. Mean amplitudes: 1.9 mV (thalamic) and 0.2–1 mV (cortical); rise times: 1.1 ms (thalamic) and 1.1–1.4 ms (cortical); half-width duration (duration from 50% of peak amplitude in rising phase to 50% of peak amplitude in falling phase): 8.8 ms for thalamic and 10.7–12.7 ms for cortical; EPSP latency jitter: 0.3 ms for thalamic and 0.7 ms for cortical. All these values can vary within a wide range. For example, retinogeniculate EPSPs can be as large as 6 mV (Bloomfield & Sherman, 1988) in the cat. In the rat visual cortex, EPSP rise times can be longer than 3 ms and EPSP durations longer than 30 ms (Mason et al. 1991).
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Correlated firing
The correlated firing generated by the N1 N2 connection is revealed by cross-correlation analysis as a monosynaptic peak displaced from zero in the correlogram (Perkel et al. 1967; Moore et al. 1970; Gerstein & Perkel, 1972; Gerstein & Aertsen, 1985; Gerstein et al. 1985). In the correlograms illustrated in this paper, the X-axis shows the time delay between N1 and N2 paired spikes (bin size = 0.1 ms for all simulated correlograms and 0.5 ms for the correlograms from Fig. 2). Positive delays (right side of the correlogram) illustrate N1 spikes that preceded N2 spikes. Negative delays (left side of the correlogram) illustrate pairs of N2 spikes preceding N1 spikes.
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A–C, three different correlograms generated by retinogeniculate connections (A, from Alonso et al. 1996), geniculocortical connections (B, from Alonso et al. 2001) and corticocortical connections (C, from Alonso & Martinez, 1998) in the cat visual pathway (bin size, 0.5 ms). The retinogeniculate correlogram was obtained by simultaneously recording from s-potentials and geniculate spikes with two different electrodes (in a serendipitous recording from two geniculate neurons that received input from the same retinal afferent). The geniculocortical correlogram was obtained in simultaneous recordings from a geniculate cell and a layer 4 cortical simple cell (area 17). The corticocortical correlogram was obtained in simultaneous recordings from a layer 4 cortical simple cell and a complex cell in layers 2 + 3 (area 17). D, average width (and S.D.) of monosynaptic peaks generated by retinogeniculate, geniculocortical and corticocortical connections. Data were taken from Usrey et al. (1999) for retinogeniculate connections (simultaneous recordings from retinal cells and geniculate cells); from Alonso et al. (2001) for geniculocortical connections; and from Alonso & Martinez (1998) for connections from layer 4 to layers 2 + 3. We chose the 10 strongest monosynaptic peaks obtained from each of these studies (10 strongest retinogeniculate peaks; 10 strongest geniculocortical peaks; 10 strongest corticocortical asymmetric peaks). The peak width was measured at half-amplitude after subtracting the baseline (equivalent to measurements at 50% of peak height in the following figures).
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A monosynaptic peak was considered to be significant if it met two criteria. First, the left side of the correlogram (between –50 and –40 ms) contained more than 50 events. Second, the amplitude of the monosynaptic peak was four S.D.s above the baseline. The correlogram baseline (mean and S.D.) was calculated by averaging all correlogram bins between –50 and –40 ms. This interval (–50, –40 ms) was ideal for baseline measurements because all our simulated correlograms were flat with the exception of the region surrounding the peak. The peak height was calculated as follows. First, we calculated the maximum of the correlogram within the interval (–10, 10 ms) and averaged the 12 bins that surrounded the bin with the highest value. Then, the peak height was measured as the average maximum minus the baseline. The peak width was measured at 25 and 50% of the peak height by using the algorithm described in eqns (4)–(7) (non-significant peaks were assigned a peak width of 0).
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where vl is the number of spikes at a given level l of the peak height (25 or 50%), b is the baseline, C is the correlogram and p is the correlogram bin with the largest amplitude. mlpre is the bin where the summation of the correlogram to the left of the peak reaches a minimum and mlpos is the bin where the summation of the correlogram to the right of the peak reaches a maximum. The peak width (wl) was defined as the number of bins between mlpre and mlpos.
We measured how the peak width changed as we modified the value of a given parameter. For each parameter value (e.g. EPSP rise time = 2 ms), we simulated 100 correlograms and calculated the average. For example, if we used 40 different values of EPSP rise time, we obtained 40 different averages. These averages were then used to calculate a peak-width ratio: maximum average/minimum average (zeros were not included in the averages). Measurements of peak-width ratio were done only for parameter values that generated at least 50% of significant correlograms. We measured the peak width at 25 and 50% peak height. We used the same approach to measure changes in the peak rise time.
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Spikes were generated in the circuit by introducing uniform noise in the membrane potential of N1. Spike collection stopped when N2 generated 2000 spikes unless specified differently in the text (e.g. for the conditions in which N2 had very low firing rates we set a time limit to stop spike collection). In initial experiments we made multiple measurements stopping spike collection at 1000, 2000, 5000, 10 000 and 50 000 N2 spikes. A value of 2000 N2 spikes was chosen because it gave us reliable measurements while allowing us to run simulations within a reasonable amount of time. Similarly, we did simulations with different firing rates for N2 and chose a range that was a good compromise between simulation speed and measurement reliability.
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Results
Strong monosynaptic connections generate a precise correlated firing between the neuron input and the neuron target. The precision of this correlated firing can be measured by cross-correlation analysis as the width of the monosynaptic peak in the correlogram. Figure 1 (top) illustrates a typical monosynaptic peak generated by a strong direct connection between N1 and N2 (see Methods for model; see also Veredas et al. 2004). Positive bins show the number of times that neuron 1 (N1) fired before neuron 2 (N2), negative bins show the number of times that N1 fired after N2, and the zero bin shows the number of times that both neurons fired together (bin width = 0.1 ms). The narrow peak displaced from zero indicates that N1 tended to fire before N2 with a time course characteristic of a monosynaptic connection. The three grey lines are from bottom to top: the baseline of the correlogram, the 25% and 50% height levels used to measure peak width. The bottom of Fig. 1 shows simulated intracellular recordings from N1 (top) and N2 (bottom). N2 is shown hyperpolarized and with no synaptic noise to make the EPSP shapes generated by N1 spikes more visible.
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Top, correlogram generated by a monosynaptic connection between two simulated neurons (N1 and N2). The correlogram indicates the number of cases that N1 fired before N2 (positive bins), after N2 (negative bins) or simultaneously with N2 (zero bin). Bin size, 0.1 ms. The correlogram shows a characteristic monosynaptic peak with a fast rise time displaced from zero indicating that N2 tended to fire after N1. The dip preceding the monosynaptic peak (left side of zero) is characteristic of correlograms generated by strong monosynaptic connections. Its shape matches the shape of the autocorrelogram from the presynaptic cell (Moore et al. 1970; see also Fig. 2). In our stimulations, the width and amplitude of this dip could be modified by increasing the ratio of the EPSP/noise amplitude and/or by increasing the refractory period. The precision of the correlated firing generated by the monosynaptic connection was estimated by measuring the width of this monosynaptic peak. The three lines on the left show the baseline (bottom) and the two levels used to measure peak width (25 and 50% of peak height). Total number of N2 spikes: 50 000. Bottom, simulated intracellular recordings of N1 (top) and N2 (bottom). N2 has been hyperpolarized and no synaptic noise has been added to make more visible the shape of the EPSPs generated by N1.
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Monosynaptic peaks are usually identified in a correlogram by their narrow width and fast rise time (Perkel et al. 1967; Levick et al. 1972; Toyama et al. 1981; Tanaka, 1983; Mastronarde, 1987; Reid & Alonso, 1995; Swadlow, 1995; Alonso & Martinez, 1998; Usrey et al. 1999; Miller et al. 2001; Roy & Alloway, 2001). However, monosynaptic connections range widely in spike-timing precision (e.g. compare Alonso & Martinez, 1998; Usrey et al. 1999; Alonso et al. 2001; compare Bloomfield & Sherman, 1988; Mason et al. 1991). In fact, it is sometimes possible to determine whether a correlogram is generated by a retinogeniculate connection or a cortical connection (layer 4 layers 2 + 3) just based on the width of the monosynaptic peak. Figure 2A, B and C shows examples of the strongest monosynaptic peaks found in our physiological experiments at each level of the visual pathway: retina thalamus, thalamus cortical layer 4, layer 4 to layer 2 + 3 ((Alonso et al. 1996, 2001; Alonso & Martinez, 1998); the retinogeniculate correlogram was obtained from simultaneous recordings of s-potentials and geniculate spikes (S-potentials are large synaptic potentials, extracellularly recorded in geniculate cells, that are generated by single retinal afferents; see Usrey et al. 1999 for simultaneous recordings from retina and thalamus). The three correlograms share many features in common. All have narrow peaks superimposed on much slower correlations (shuffle correlograms for Fig. 2B and C are shown in Fig. 3 of Alonso et al. 1996 and Fig. 2 of Alonso & Martinez, 1998, respectively). All have peaks that are displaced from zero indicating that one cell tended to fire before the other. Finally, in all cases, the left side of the correlogram has a dip that matches the autocorrelogram of the presynaptic cell as would be expected from a strong monosynaptic connection (Moore et al. 1970).
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Top, three correlograms (N1 N2) obtained with different N2-EPSP rise times. As the rise time becomes longer the monosynaptic peaks become smaller and broader. The shapes of the EPSP and EPSP derivatives (EPSPd) used in the simulations are shown at the top of the correlograms. Note that the width of the monosynaptic peak in the correlogram resembles more closely the width of the EPSPd than the width of the EPSP. Total number of N2 spikes for all three correlograms, 5000. Bottom, plot showing an increase in peak width as the EPSP rise time increases. We ran 100 simulations for each value of EPSP rise time (100 correlograms) and represented the average and S.D. at 25% of peak height (black) and 50% of peak height (grey). A circuit representing the neurons involved is shown at the left: N2 receives excitatory input from N1 in addition to random synaptic noise (see Methods). Examples of the EPSPs used in the simulations are shown on top of the circuit diagram (shown without synaptic noise; total time shown, 10 ms; EPSP amplitude, 0.76 mV). Firing rates: N1, 107 spikes s–1; N2, 51–90 spikes s–1.
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There is also a noticeable difference among the three correlograms. The peak width is narrowest in the retinogeniculate connection (Fig. 2A) and widest in the corticocortical connection (Fig. 2C). This difference in peak width can be observed in individual cases (Fig. 2A, B and C) and on averages obtained from several cases (Fig. 2D; the retinogeniculate data for Fig. 2D was generously provided by M. Usrey, J Reppas and C. Reid from Usrey et al. 1999). A similar difference in peak width is found when comparing the thalamic synchrony generated by divergent retinal afferents with the cortical synchrony generated by divergent cortical afferents (e.g. compare Alonso et al. 1996; Alonso & Martinez, 1998).
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Computer simulations were used to identify the main factors that determine these differences in correlated firing precision. We used the simplest circuit possible – two neurons that are monosynaptically connected – and modified each parameter separately to study its contribution to correlated firing. Throughout the entire paper we measure the precision of the correlated firing as the width and rise time of the monosynaptic peak in the correlogram.
EPSP rise time
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In a classical paper, Fetz & Gustafsson (1983) showed that the precision of the correlated firing generated by a monosynaptic connection was strongly determined by the rise time of the EPSP and matched the EPSP derivative. EPSPs with fast rise times generated narrow monosynaptic peaks and EPSPs with slow rise times generated broader peaks (see also Matsumura et al. 1996). In contrast, the monosynaptic correlograms were found to resemble more closely the shape of the EPSP than the EPSP's derivative in Aplysia neurons (Moore et al. 1970) and the motoneurons of the crayfish claw (Lindsey & Gerstein, 1979).
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Measurements of EPSP rise time in experimental studies are usually obtained from averages and therefore they reflect the combination of two different variables: EPSP rise time and EPSP latency jitter (see below). Similarly, previous simulation studies (Surmeier & Weinberg, 1985) modified two parameters at the same time (EPSP rise time and EPSP amplitude) making it difficult to determine the contribution of the EPSP rise time in isolation. Our simulations allowed us to re-examine this important issue by generating a large number of correlograms using EPSPs that differed mainly in their rise times. We measured the width of the monosynaptic peak obtained with different values of EPSP rise time (Fig. 3). Examples of correlograms are shown at the top of the figure and the measurements for all correlograms at the bottom. Increasing the EPSP rise time had two main effects on the correlated firing generated by the connection N1 N2. First, it reduced the amplitude of the monosynaptic peak, and second, consistently with the findings of Fetz & Gustafsson (1983), it increased the width of the monosynaptic peak. Also consistent with Fetz & Gustafsson (1983), the width of the monosynaptic peak matched better the width of the EPSP derivative (EPSPd) than the width of the EPSP (illustrated at the top of the correlograms). We quantified the changes in peak width by calculating the average peak width for each value of rise time and then dividing the maximum by the minimum average (see Methods). The peak width ratio measured by this method was 5.53 (at 25% peak height) and 2.18 (at 50% peak height) and there was a strong correlation between EPSP rise time and peak width (0.74 and 0.91, respectively, Table 1).
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These results are in close agreement with Fetz & Gustafsson (1983; see also Knox, 1974) and extend previous results from Surmeier & Weinberg (1985) by showing that changes in EPSP rise time can modify the shape of the monosynaptic correlogram even if the EPSP duration remains relatively unchanged.
EPSP duration
It is widely believed that the EPSP duration has a small influence on the correlated firing generated by a monosynaptic connection (e.g. Kirkwood, 1979; Fetz & Gustafsson, 1983; Reid, 2001). This belief probably arises from previous experimental studies (Fetz & Gustafsson, 1983; Matsumura et al. 1996) and from the logical assumption that the EPSP decay is below threshold and should have a minor contribution to neuronal firing. Our results indicate quite the opposite. The EPSP duration may be one of the most important parameters in determining the correlated firing precision of some neurons. Figure 4 shows the results obtained by calculating multiple N1 N2 correlograms with different EPSP durations. EPSPs with short durations generated correlograms that resembled very closely those generated by retinogeniculate connections. In contrast, EPSPs with longer durations generated monosynaptic peaks with very long tails more typical of corticocortical connections. Long EPSPs hold the synaptic activity near threshold for long periods of time making the monosynaptic peaks broad. In the extreme case, if the EPSP is too long and the firing rate of N1 too high, N2 fires at its maximum capacity and the precise correlated firing between N1 and N2 fades away – the firing of N2 no longer depends on N1. This extreme situation is likely to be avoided in physiological circuits by disynaptic inhibition (Pouille & Scanziani, 2001; Wehr & Zador, 2003; Kuhn et al. 2004), sequences of inward–outward currents recruited by strong stimuli (Axmacher & Miles, 2004) or other mechanisms (e.g. synaptic depression). From all the parameters that we have studied, the EPSP duration was among the ones that generated the most pronounced changes in the width of the N1 N2 monosynaptic peak (width ratio at 25% was 11.87, Table 1). These results are consistent with data showing that action potential timing is more variable in neurons that generate long EPSPs than those generating short EPSPs (Fricker & Miles, 2000; see also Pouille & Scanziani, 2001; Wehr & Zador, 2003).
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Correlograms (N1 N2) obtained with different N2-EPSP durations. Increasing the EPSP duration made monosynaptic peaks broader. There was also an increase in the correlogram baseline due to the more effective temporal summation that led to a higher firing rate in N2. Notice the difference in the scale of the Y-axis (when EPSPs are brief most N2 spikes are precisely correlated with N1 spikes). As in Fig. 3, the shapes of the EPSP and EPSP derivatives (EPSPd) are shown at the top of the correlograms. Unlike in Fig. 3, the width of the monosynaptic peak in the correlogram resembles more closely the width of the EPSP than the width of the EPSPd. A cartoon of the circuit and examples of the EPSPs used are shown on the bottom left (shown without synaptic noise; total time shown, 80 ms; EPSP amplitude, 3.85 mV). Symbols as in Fig. 3. Firing rates: N1, 25 spikes s–1; N2, 15–175 spikes s–1.
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Threshold
Another parameter that could influence the precision of the correlated firing generated by a monosynaptic connection is the neuronal threshold. Neurons that have their baseline activity close to threshold fire more and could potentially generate broader correlograms because the EPSP is above threshold for a longer period of time (Kirkwood, 1979). As shown in Fig. 5, making the neuronal threshold higher (less negative) did not cause pronounced changes in peak width (see also Table 1). As expected, N2 became less excitable and there was a reduction in the number of events in the correlogram due to a reduction in the firing rate of N2. When the threshold was too high (e.g. –36 mV) N2 did not fire and the correlogram was non-significant (peak width = 0); when the threshold was too low (e.g. –41 mV) N2 fired at its maximum capacity totally independent from N1 (flat correlogram). Within the ranges that generated significant correlograms, changes in threshold had a relatively small influence on the width of the monosynaptic peak (Table 1). It should be noted, however, that these results were obtained using a static threshold and may differ from results obtained with thresholds that are more dynamic (Azouz & Gray, 2000, 2003).
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Correlograms (N1 N2) obtained with different N2 thresholds. Increasing the threshold generated only modest changes in the width of the monosynaptic peaks. The changes in the correlogram baseline are due to changes in the firing rate of N2. A cartoon of the circuit and examples of the EPSP used is shown on the bottom left (shown as spike trigger average; total time shown, 20 ms; EPSP amplitude, 2.2 mV). Firing rates: N1, 100 spikes s–1; N2, 165–35 spikes s–1.
Synaptic noise (amplitude and time course)
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The precision of the correlated firing generated by a monosynaptic connection (N1 N2) could be determined by the amplitude and time course of the multiple synaptic events that do not originate in N1. To test this possibility, we used a gaussian distribution of synaptic-event amplitudes (synaptic noise) and modified the S.D. of the gaussian while keeping the average current constant. For a S.D. of 0, the synaptic noise was a constant current and, as we increased the S.D., synaptic events of increasingly larger amplitude appeared in the baseline. Increasing the S.D. of the synaptic-noise distribution made monosynaptic peaks smaller because more N2 spikes were generated by synaptic events that did not originate in N1 (Fig. 6). In addition, the monosynaptic peaks became slightly wider (Table 1).
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In a different experiment, we modified the time course of the synaptic noise while keeping constant the average amplitude. In these simulations, the synaptic noise was convolved with exponential functions (e–t/) with distinct time constants () to generate noise with different frequency spectra (re-scaling the amplitude of the noise after filtering). These simulations were not able to generate large changes in the width of the monosynaptic peak (Fig. 7, Table 1; similar results were obtained by sampling the noise with different frequencies and reconstructing it with a cubic spline).
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Ratio of EPSP/synaptic noise
One of the most significant differences between retinogeniculate connections and corticocortical connections is in their synaptic strength. While a single retinal afferent can make more than one hundred synapses with a geniculate neuron (Hamos et al. 1987), the number of synapses per axon in geniculocortical and corticocortical connections is usually smaller than 10 (e.g. Freund et al. 1985; Feldmeyer et al. 2002). It is possible that differences in the precision of correlated firing between retinogeniculate connections and corticocortical connections are due to differences in the relative amplitude of the monosynaptic EPSPs. We investigated this possibility by modifying the relative amplitude of the postsynaptic EPSP with respect to the synaptic noise while keeping the mean firing rate of N2 constant.
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The results of these simulations indicate that the amplitude of the monosynaptic EPSP is strongly related with the amplitude of the monosynaptic peak in the correlogram but much less with its width or precision (see also Poliakov et al. 1997). Figure 8 shows simulated correlograms that were obtained with different EPSP/noise amplitude ratios. Increasing the ratio of the signal to noise had a pronounced effect on the amplitude of the monosynaptic peak but barely affected the peak width (Table 1).
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Correlograms obtained with three different ratios of EPSP/synaptic noise. For most EPSP/noise ratios, increments in the ratio did not generate pronounced changes in peak width. Only the smallest ratios (EPSP/noise < 1) generated slightly broader peaks. On the left, examples of EPSPs + synaptic noise used (shown as spike trigger averages; total time shown, 20 ms; Amplitude of EPSP at the bottom, 8 mV). Symbols as in Fig. 3. Firing rates: N1, 75 spikes s–1; N2, 110 spikes s–1.
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EPSP latency jitter
Monosynaptic connections also generate latency jitter, that is, the time delay between a presynaptic spike and the postsynaptic EPSP varies. The amount of temporal jitter depends on the type of neurons involved (Markram et al. 1997a; Feldmeyer & Sakmann, 2000; Feldmeyer et al. 2002). For example, in the developing rat neocortex, Markram et al. (1997a) measured fluctuations in EPSP latency of 1.5 ms between layer 5 neurons which is a very large value in comparison with the submillisecond jitter produced by retinogeniculate connections (Cleland et al. 1971a,b; Eysel, 1976; So & Shapley, 1979; Bloomfield & Sherman, 1988). The EPSP latency jitter reaches its highest values in polysynaptic connections and it is usually used as a criterion to distinguish between monosynaptic and disynaptic inputs (e.g. Ferster & Lindstrom, 1983). Therefore, to understand the role of polysynaptic inputs in generating precise correlated firing it is important to study how the polysynaptic correlated firing is affected by the total EPSP latency jitter (the total EPSP latency jitter depends on the jitter of each monosynaptic connection in the polysynaptic chain). Here, we studied the effect of increasing the latency jitter on the correlated firing generated by the connection N1 N2. Figure 9 (top) shows examples of correlograms obtained with three different jitter values. As we increased the latency jitter the monosynaptic peak became smaller and wider because many postsynaptic spikes were no longer precisely correlated with their presynaptic spikes (reduction in peak amplitude) and became distributed over longer periods of time (increase in the width of the peak tail). Changes in EPSP latency jitter also generated pronounced changes in the rise time of the monosynaptic peak (Table 1). As shown in Fig. 9, the connection N1 N2 frequently failed to generate precise correlated firing for jitter values larger than 1.6 ms. This result is important because it suggests that a polysynaptic connection has to generate very small latency jitter to generate correlograms that resemble monosynaptic connections. Typical polysynaptic correlogram peaks can be easily generated with large values of EPSP duration and latency jitter (Fig. 10, Table 1).
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Correlograms obtained with three different EPSP latency jitters (measured as the S.D. of a gaussian distribution for synaptic delay). Small increments in jitter produced a strong reduction in the amplitude of the monosynaptic peak and an increase in peak width. We could not generate significant correlograms when the latency jitter was larger than 1.6 ms. On the left cartoon of the circuit and examples of EPSPs used (total time shown, 40 ms; EPSP amplitude, 3.85 mV). Symbols as in Fig. 3. Firing rates: N1, 100 spikes s–1; N2, 150 spikes s–1.
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Correlograms obtained while increasing both the latency jitter and EPSP duration. As the latency jitter and the EPSP duration increases the monosynaptic peaks become weaker and broader. On the left, cartoon of the circuit and examples of EPSPs used (shown without synaptic noise; total time shown, 40 ms; EPSP amplitudes, 5.85 mV). Symbols as in Fig. 3. Firing rates: N1, 75 spikes s–1; N2, 44–330 spikes s–1.
Interactions between pairs of parameters
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The results presented above indicate that the width of a monosynaptic peak in a correlogram is mostly determined by the EPSP time course (rise time, duration and latency jitter). However, any conclusion from these results is limited by the fact that we did not explore all possible parameter combinations. In our simulations each parameter was modified independently while keeping the other parameters at default values. For example, to study how the width of the monosynaptic peak depended on EPSP rise time, we modified the value of EPSP rise time while keeping the EPSP duration at a default value of 8.7 ms. While this strategy allowed us to examine the contribution of each factor independently, the results might be different if we had used other default values. To address this limitation we repeated all the simulations described above using three different default values for each pair combination of parameters tested (see Methods for details on the default values used). For example, we studied how peak width changes with EPSP rise time (test parameter) and EPSP duration (default parameter) by running the simulations for EPSP rise time of Fig. 3 three times, each time using a different value of EPSP duration. The largest peak-width ratio obtained in the three simulations was plotted as a circle in an X–Y grid (Fig. 11A; the circle diameter represents the value of the ratio).
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Changes in peak width (A and C) and peak rise time (B and D) obtained with different pair combinations of parameters (measured at 25% of peak height). A, the circle diameter represents the value of the peak-width ratio. The circle with a number indicates the largest ratio measured within the entire grid (e.g. the largest peak-width ratio was 22). The peak-width ratio for each pair was obtained as follows. We changed the value of a specific parameter (test parameter) and obtained a peak-width ratio (maximum peak width/minimum peak width). We repeated the measurements three times using three different values of a specific default parameter (default parameter) and obtained three peak-width ratios. The maximum ratio obtained in the three measurements was selected and shown in this figure. For example, the circle with the number 22 (test parameter, EPSP rise time; default parameter, EPSP duration) was obtained by running simulations for EPSP rise time three times using EPSP duration of 2.4, 14.4 or 30.3 ms. The diagonal of the plot shows the results from Figs 3, 4, 5, 6, 8 and 9 obtained using only one default value (e.g. EPSP rise time/EPSP rise time, EPSPd = 8.7 ms). B, same as A, but measuring peak rise at 25% of peak height. C, the circle diameter represents the correlation index between peak width and the test parameter. The circle with a number (0.97) indicates the strongest correlation measured within the entire grid (e.g. the strongest correlation was obtained in the simulations for EPSP duration using an EPSP/noise ratio of 5.26). The correlation index for each pair of test parameter/default parameter was obtained as follows. We measured the correlogram peak widths obtained with multiple values of a test parameter (as in Figs 3–9) and obtained a correlation index between peak width and the test parameter. This measurement was repeated three times using three different values of a specified default parameter. The diagonal of the plot shows the results from Figs 3,4,5,6, 8 and 9 obtained using only one default value. D, same as C, but measuring peak rise at 25% of peak height. EPSPr, EPSP rise time; EPSPd, EPSP duration; Thres, threshold; Noise, synaptic noise; EPSP/n, ratio of EPSP/synaptic noise amplitudes; Jitter, EPSP latency jitter.
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The largest peak-width ratio within the entire grid (peak-width ratio = 22) was obtained in the simulations for EPSP rise time (test parameter) while using a EPSP duration of 30.3 ms. The simulations for EPSP duration and latency jitter also yielded large peak-width ratios. Figure 11B shows similar paired measurements for peak rise. The largest peak rise within the entire grid (peak-rise ratio = 16) was obtained by changing the latency jitter in the absence of synaptic noise (default value for noise = 0 nA). Finally, in most simulations the values of the test parameter were highly correlated with the values of peak width (Fig. 11C) and peak rise (Fig. 11D). The results from these paired simulations are consistent with our previous conclusion: the EPSP time course is the most important factor in determining the width of the monosynaptic peak.
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Simulations of monosynaptic correlograms in the absence of synaptic noise
The synaptic noise was a very important parameter in our simulations; without noise it was usually difficult to generate physiological correlograms. Figure 12 shows the results from a simulation in which we modulated the membrane potential of N2 with a ramp instead of noise (amplitude, 0.0025–0.005 nA; frequency, 33 Hz). In these simulations N2 generated bursts of spikes that were extremely precise in timing due to the lack of random fluctuations in the membrane potential. As a consequence of this artificially precise spike timing, the N1 N2 monosynaptic correlogram had multiple peaks that reproduced the burst pattern of N2. These multiple peaks are not found in physiological experiments probably because they are filtered by the synaptic noise. If gaussian noise is added to the simulations, the multiple peaks from the postsynaptic bursts fuse into a single peak. Consistently with physiological data, the monosynaptic peak is wider when the postsynaptic cell fire bursts instead of single spikes (Creutzfeldt et al. 1980; Gray et al. 1990; Eggermont et al. 1993). Figure 13 shows the results from simulations using a ramp to modulate the membrane potential of N2, but in this case N2 fired mostly single spikes (in these simulations N2 received monosynapic excitation and disynaptic inhibition from N1). In these simulations, the lack of bursts together with the disynaptic inhibition made the correlograms peak narrower than in the previous figure (the number of multiple peaks is reduced). Regardless of the differences in the shape of the correlograms, the results from the simulations using a ramp baseline were equivalent to the results using gaussian noise – increasing the EPSP rise time made monosynaptic peaks wider (notice that the measurements of peak width are mostly determined by the first peak of the correlogram; see Methods). Similar results were obtained by modulating the membrane potential of N2 with a sinusoid instead of a ramp.
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Correlograms obtained with different values of EPSP rise time (the membrane potential of N2 was modulated with a ramp instead of gaussian noise). Notice the multiple peaks in the correlogram due to the lack of variability in the membrane potential. Consistently with the results shown in Fig. 3, the monosynaptic peak closest to zero becomes weaker and slightly broader when the EPSP rise time increases. The amplitude of the ramp was 0.0025–0.005 nA and the frequency was 33 Hz. On the left, cartoon of the circuit and examples of EPSPs used (total time shown, 10 ms; EPSP amplitudes, 0.76 mV). Symbols as in Fig. 3. Firing rates: N1, 107 spikes s–1; N2, 128–153 spikes s–1.
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Correlograms obtained with different values of EPSP rise time (the membrane potential of N2 was modulated with a ramp instead of gaussian noise). N2 fired mostly single spikes and received both monosynaptic excitation and disynaptic inhibition from N1 (the disynaptic inhibition generates a dip on the right side of the correlogram). Consistently with the results from Fig. 3, the monosynaptic peak becomes weaker and broader as the EPSP rise time increases. On the left, cartoon of the circuit and examples of EPSPs used (total time shown, 10 ms; EPSP amplitudes, 0.76 mV). Symbols as in Fig. 3. Firing rates: N1, 107 spikes s–1; N2, 13–16 spikes s–1.
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While we were able to study the influence of EPSP rise time on the width of the monosynaptic peak using ramp/sinusoids as baseline, this approach did not work for EPSP duration. As the EPSP duration increased, temporal summation became so effective that N2 fired at its maximum capacity totally independent from N1. Without synaptic noise, increasing the EPSP duration on top of a ramp/sinusoid invariably led to a flat correlogram unless we modified several parameters at the same time (e.g. increase EPSP duration and reduce the firing rate of N1; increase EPSP duration and reduce the amplitude of the ramp). In other words, it was not possible to find a default parameter range that would work for all the EPSP durations when using a ramp/sinusoid instead of synaptic noise as the baseline. This problem could be partially solved by adding disynaptic inhibition to the circuit, but even then, the correlogram became flat for EPSP durations longer than 14 ms (Fig. 14).
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Correlograms obtained using different values of EPSP duration. In addition to monosynaptic excitation, N2 receives disynaptic inhibition from N1 and its membrane potential is modulated by a ramp instead of noise. Without synaptic noise, it was not possible to investigate the influence of EPSP duration on the shape of the monosynaptic peak. For EPSPs longer than 14 ms, the temporal summation was so effective that N2 fired at its maximum capacity, independently of N1. (If the parameters of the model were adjusted to reduce the temporal summation of the long EPSPs, shorter EPSPs were not able to make N2 fire). Therefore, when using a ramp instead of noise we cannot measure how changes in EPSP duration only, affect the shape of the monosynaptic correlogram. On the left, cartoon of the circuit and examples of EPSPs used (total time shown, 80 ms; EPSP amplitudes, 3.85 mV). Symbols as in Fig. 3. Firing rates: N1, 107 spikes s–1; N2, 2–254 spikes s–1.
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Discussion
The precision of the correlated firing generated by a monosynaptic connection varies across the visual pathway. It is highest in retinogeniculate connections and lower in geniculocortical and corticocortical (layer 4 to layers 2 + 3) connections. Here we have examined some of the factors that may be responsible for these differences in precision. Among all the parameters studied, the EPSP rise time, duration and latency jitter generated the most pronounced changes in the width of the monosynaptic peaks. This result is in close agreement with previous studies in cat motoneurons (Kirkwood & Sears, 1982; Fetz & Gustafsson, 1983; Gustafsson & McCrea, 1984; Cope et al. 1987; Poliakov et al. 1997), Aplysia neurons (Moore et al. 1970), the motoneurons of the crayfish claw (Lindsey & Gerstein, 1979) and more recent data from GABAergic neurons in neocortex (Galarreta & Hestrin, 2001). The result is also consistent with recent studies in the hippocampus showing that spike timing precision is greatly influenced by the shape of the EPSP, which can be prolonged by voltage-gated conductances or shorten by disynaptic inhibition (Fricker & Miles, 2000; Pouille & Scanziani, 2001; Axmacher & Miles, 2004).
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EPSPs generated by corticocortical connections tend to be longer in duration than the EPSPs generated by retinogeniculate connections and have more latency jitter (Eysel, 1976; Bloomfield & Sherman, 1988; Stratford et al. 1996; see Methods for detail). Therefore, the EPSP duration and latency jitter could be the main factors that determine the differences in correlated firing precision at different levels of the visual pathway. The differences in EPSP duration between thalamic and cortical neurons could be due to differences in the properties of the cell membrane (e.g. input resistance) and/or differences in the voltage-gated synaptic conductances that shape the EPSPs (Reyes & Fetz, 1993; Stuart & Sakmann, 1995; Andreasen & Lambert, 1999; Fricker & Miles, 2000). In the hippocampus, the EPSPs from pyramidal cells are greatly prolonged by the activation of a Na+ current; however, the EPSPs from inhibitory neurons show little voltage-dependent amplification and are shorter in duration (Fricker & Miles, 2000). The LGN and visual cortex could behave like inhibitory and pyramidal neurons in hippocampus; the EPSPs from LGN neurons might be less amplified by voltage-gated conductances than the EPSPs of cortical neurons. However, our model does not allow us to address this issue.
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Like with any model, ours is not free of limitations. By choosing an IAF model, we chose not to study important cellular properties that influence the shape of the EPSP. The EPSP is influenced by multiple factors that include membrane properties (e.g. time constant, input resistance), voltage-gated conductances, synaptic depression and disynaptic inhibition. The choice of the neuronal model was based on two criteria. First, we chose a model that would allow us to generate realistic monosynaptic correlograms (that resemble the physiological correlograms measured at the different stages of the visual pathway). Second, we chose a model that would allow us to generate thousands of correlograms in a reasonable amount of time (4000 correlograms per day). A crucial parameter in our model was the synaptic noise. In our hands, it was very difficult to simulate realistic monosynaptic peaks without using some type of synaptic noise (see Fig. 12). The gaussian synaptic noise simulates quite well the fluctuations observed in the membrane potential of intracellularly recorded neurons. In vitro intracellular recordings consistently show that monosynaptic EPSPs are embedded in gaussian noise (synaptic events with normally distributed amplitudes e.g. Stratford et al. 1996; Feldmeyer et al. 2005). The amplitude of this noise is clearly underestimated by the measurements in vitro since most of the connections are cut in this preparation; in vivo recordings show much noisier synaptic baselines (e.g. Hirsch et al. 1998).
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It should be emphasized that correlograms are shaped by other factors that were not explored here such as the stimulus conditions and changes in excitability (Perkel et al. 1967; Lee et al. 1977; Palm et al. 1988; Aertsen et al. 1989; Brody, 1998; Dorn & Ringach, 2003). However, it is important to notice that correlations generated by direct connections tend to be narrower than those generated simply by the stimulus (unless the stimulus is rapidly modulated and spatially diffuse; see Reid, 2001 for review). As shown here, most polysynaptic inputs should fail to generate precise correlated firing unless they generate very small values of EPSP latency jitter. In our simulations, monosynaptic peaks were very weak when the EPSP jitter was larger than 1.6 ms. Since most disynaptic connections generate jitter values larger than 2 ms (see below), the contribution of disynaptic inputs to the precise correlated firing generated by monosynaptic connections should be relatively small. Consistently, in cat visual cortex, the correlograms generated by disynaptically connected geniculate cells and layers 2+ 3 cells do not show significant monosynaptic peaks (Alonso & Martinez, 1998). Exceptionally, retinocortical disynaptic connections (retina LGN cortex) are able to generate precise correlated firing (Lee et al. 1977; Kara & Reid, 1999) probably due to the very small latency jitter of retinogeniculate connections (Cleland et al. 1971a, b; Eysel, 1976; So & Shapley, 1979; Mastronarde, 1987; Bloomfield & Sherman, 1988; Usrey et al. 1999).
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The argument that most polysynaptic inputs contribute little to the correlated firing generated by monosynaptic connections relies heavily on the values of latency jitter measured in physiological experiments. Recent studies using dual intracellular recordings combined with anatomical reconstructions are finding values of EPSP latency jitter of more than 1 ms in cortical neurons (e.g. Markram et al. 1997a; Feldmeyer & Sakmann, 2000; Feldmeyer et al. 2002). In addition, the jitter between the EPSP onset and the generation of a spike can be as large as 0.8 ms for geniculocortical connections in vivo which yields a total effective latency jitter of 1.8 ms for some monosynaptic connections in the cortex (Ferster & Lindstrom, 1983). Based on these estimates, a disynaptic connection should generate latency jitter values at least twice as large (3.6 ms). However, 3.6 ms is likely to be an underestimate; some data indicate that this value could be larger than 10 ms (Stuart & Sakmann, 1995).
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The EPSP duration varies among cortical neurons; consequently, not all corticocortical connections generate broad correlograms and not all corticocortical EPSPs have long durations. For example, inhibitory neurons in layer 4 of the somatosensory cortex can fire in very precise synchrony (Swadlow et al. 1998) probably due to strong geniculocortical connections (Swadlow, 1995; Swadlow et al. 2002) and electrical junctions (Galarreta & Hestrin, 1999; Gibson et al. 1999; Galarreta & Hestrin, 2001).
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It should be emphasized that our results are in good agreement with those from Fetz & Gustafsson (1983). Like Fetz & Gustafsson (1983), we find that the precision of the correlated firing generated by monosynaptic connections depends on the rise time of the EPSP: EPSPs with slow rise times generate broader monosynaptic peaks than EPSPs with fast rise times. Our results indicate, however, that the EPSP duration is also an important factor, a conclusion that is more in agreement with Moore et al. (1970), Lindsey & Gerstein (1979) and recent intracellular studies (Fricker & Miles, 2000; Pouille & Scanziani, 2001; Wehr & Zador, 2003; Axmacher & Miles, 2004). The differences in the results could be explained by differences in the EPSPs durations of the neurons studied. The width of the monosynaptic peaks studied by Fetz & Gustafsson (1983) ‘depended mostly on the EPSP rise time’ probably because the EPSPs of cat motoneurons are very short in duration. In contrast, the width of the monosynaptic peaks studied by Moore et al. (1970) depended mostly on the EPSP duration because the EPSPs recorded in Aplysia cells are very long (total duration > 200 ms as shown in Fig. 2 of Moore et al. 1970). It is important to remember that EPSP rise time and decay time are strongly correlated: EPSPs with slow rise times are usually longer in duration than EPSPs with fast rise times. Therefore, the results of Moore et al. (1970), Lindsey & Gerstein (1979) and Fetz & Gustafsson (1983) may not be so different after all.
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The finding that the correlated firing of monosynaptic connections could be strongly determined by the EPSP time course provides support for neural codes based on precise spike timing (Mainen & Sejnowski, 1995; Singer & Gray, 1995; Nowak et al. 1997; Reich et al. 1997; Diesmann et al. 1999; Reinagel & Reid, 2000). EPSPs with long durations may reduce spike-timing precision but they also increase the chances for two inputs to temporally interact. Therefore, the loss in synchrony precision from thalamus to visual cortex should not preclude the visual system to use spike timing for visual processing.
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2 Department of Psychology, University of Connecticut, Storrs, CT, USA
3 SUNY State College of Optometry, New York, NY, USA
Abstract
Across the visual pathway, strong monosynaptic connections generate a precise correlated firing between presynaptic and postsynaptic neurons. The precision of this correlated firing is not the same within thalamus and visual cortex. While retinogeniculate connections generate a very narrow peak in the correlogram (peak width < 1 ms), the peaks generated by geniculocortical and corticocortical connections have usually a time course of several milliseconds. Several factors could explain these differences in timing precision such as the amplitude of the monosynaptic EPSP (excitatory postsynaptic potential), its time course or the contribution of polysynaptic inputs. While it is difficult to isolate the contribution of each factor in physiological experiments, a first approximation can be done in modelling studies. Here, we simulated two monosynaptically connected neurons to measure changes in their correlated firing as we independently modified different parameters of the connection. Our results suggest that the precision of the correlated firing generated by strong monosynaptic connections is mostly determined by the EPSP time course of the connection and much less by other factors. In addition, we show that a polysynaptic pathway is unlikely to emulate the correlated firing generated by a monosynaptic connection unless it generates EPSPs with very small latency jitter.
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Introduction
Neuronal response properties undergo several important transformations from the retina to the primary visual cortex. Receptive fields become more elaborated (Hubel & Wiesel, 1962), average firing rates are reduced and visual responses become more variable (Kara et al. 2000). Studies of cross-correlation analysis suggest an additional transformation – the correlated firing between presynaptic and postsynaptic neurons becomes less precise. While retinogeniculate connections generate very narrow correlograms with less than 1 ms width at half-amplitude (Levick et al. 1972; Mastronarde, 1987; Usrey et al. 1999), the connections from the lateral geniculate nucleus (LGN) to cortical layer 4 and from layer 4 to layers 2 and 3 generate much wider correlograms (Toyama et al. 1981; Tanaka, 1983; Reid & Alonso, 1995; Alonso & Martinez, 1998; Alonso et al. 2001).
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It is unclear what factors are responsible for these differences in correlated firing precision. Retinogeniculate connections differ from geniculocortical and corticocortical connections in a large variety of anatomical and physiological factors that could be important in determining precision. While a retinal afferent makes more than a hundred synapses with a single geniculate cell (Hamos et al. 1987), a geniculate or a cortical afferent makes usually less than 10 synapses per neuronal target (Freund et al. 1985; Feldmeyer et al. 2002). Retinogeniculate EPSPs (excitatory postsynaptic potentials) are also larger, faster, shorter and have less jitter in their latency than corticocortical EPSPs (e.g. Eysel, 1976; Bloomfield & Sherman, 1988; Stratford et al. 1996). Moreover, while one to three retinal afferents converge into a single geniculate neuron, a layer 4 neuron may receive convergent input from around 30 geniculate afferents (Cleland et al. 1971a; Chen & Regehr, 2000; Alonso et al. 2001). Cortical neurons are also heavily interconnected while geniculate cells barely make excitatory connections with each other (Friedlander et al. 1979).
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Previous studies in the motor and somatosensory systems did not reach an agreement on whether the width of a monosynaptic peak in a correlogram is determined by the EPSP rise time (Knox, 1974; Fetz & Gustafsson, 1983), the EPSP duration (Moore et al. 1970; Lindsey & Gerstein, 1979; Surmeier & Weinberg, 1985) or a linear combination of both (Kirkwood & Sears, 1982; Gustafsson & McCrea, 1984; Cope et al. 1987). Moreover, although the contribution of other parameters such as the synaptic noise and threshold were thought to have an important contribution, they were not explored in detail (see Kirkwood, 1979 for review). Here, we have systematically explored the contribution of a large number of synaptic parameters that could determine the width of a monosynaptic correlogram in the visual pathway. In contrast to previous studies (Surmeier & Weinberg, 1985), we were able to modify different parameters independently (e.g. EPSP rise time and duration). Moreover, our simulations used parameter values consistent with synaptic physiology measurements (Eysel, 1976; Bloomfield & Sherman, 1988; Stratford et al. 1996) and generated realistic correlograms that resemble very closely those measured at three different stages of the visual pathway (retina–LGN (Levick et al. 1972; Mastronarde, 1987; Usrey et al. 1999); LGN–cortex (Tanaka, 1983; Reid & Alonso, 1995; Alonso et al. 2001); cortical layer 4–superficial layers of cortex (Alonso & Martinez, 1998)).
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The precise correlated firing generated by presynaptic and postsynaptic neurons plays a major role in the development and plastic remodelling of the visual pathway (Katz & Shatz, 1996) since the relative timing of discharges is a crucial parameter (Bienenstock et al. 1982; Bell et al. 1997; Markram et al. 1997b; Bi & Poo, 1998; Debanne et al. 1998; Song et al. 2000; Sjostrom et al. 2001; Froemke & Dan, 2002; Fu et al. 2002). Therefore, it is important to identify the factors that are responsible for the differences in correlated-firing precision. Our results suggest that the most important factors are the EPSP time course (rise time, duration and latency jitter). EPSPs with large values of duration and latency jitter generate broad correlogram peaks and those with small values generate narrow correlogram peaks. This finding resonates quite well with recent results in the hippocampus, which indicate that spike-timing precision is strongly influenced by the shape of the compound EPSP (Fricker & Miles, 2000; Pouille & Scanziani, 2001; Axmacher & Miles, 2004; see also Poliakov et al. 1997). For example, CA1 inhibitory neurons, which generate short-duration EPSPs, initiate most of their spikes at the rising phase of the EPSP. In contrast, CA1 pyramidal neurons, which generate prolonged EPSPs (due to the activation of a Na+ current) initiate their spikes both at the rising and decay phase of the EPSP (Fricker & Miles, 2000). Spikes are also most frequently generated at the rising phase of the EPSP when the compound EPSP is shorten by strong sequences of outward–inward currents (Axmacher & Miles, 2004) or strong IPSPs (Pouille & Scanziani, 2001; Wehr & Zador, 2003). Based on these experimental data and our results, we propose that the EPSP time course increases from thalamus to visual cortex, reducing the precision of the monosynaptic correlated firing while making temporal summation more effective.
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Methods
Throughout this paper we use two simulated neurons that are monosynaptically connected: N1 is the presynaptic neuron and N2 is the postsynaptic neuron. In addition to the input from N1, synaptic noise is injected in the membrane potential of N2 to simulate the contribution of other inputs that are not N1. Synaptic noise is also injected in N1 to generate presynaptic spikes. The monosynaptic connection generates a precise correlated firing that we measure with cross-correlation analysis. The precision of the correlated firing is estimated from the width of the monosynaptic peak in the correlogram. We measure how the peak width is modified as we change different parameters of the monosynaptic connection. In addition, we measure changes in the rise time of the monosynaptic peak, which are summarized in Table 1.
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The neuron model
Compartmental models take into account the geometrical and physiological characteristics of the neuron, like axons, dendritic spines, ionic channels or soma. On the other hand, point models characterize the neuron's electrical behaviour, and consider it as a single point in space (Hodgkin & Huxley, 1952; FitzHugh, 1961; Nagumo et al. 1962; Morris & Lecar, 1981). These simplified models can introduce the main features of neuronal physiology with a relatively low computational cost. Integrate-and-fire (IAF) neurons are a particular case of simplified models that derive from the pioneering work of Louis Lapicque (Lapicque, 1907, 1926). Different versions of this model have been proposed in various studies (Stein, 1967a,b; Knight, 1972; Jack et al. 1983; Tuckwell, 1988). The traditional form of an IAF model is a first-order differential equation (eqn (1)) whose dynamics are characterized by a subthreshold integration domain (where the neuron integrates the inputs and behaves like a RC circuit) and a threshold voltage for the generation of action potentials.
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In eqn (1), Rm and Cm are the neuronal membrane resistance and capacitance, respectively, Vm is the membrane potential, Vrest is the resting potential and I(t) is the input current. In eqn (2), the input current I(t) is determined by the neuronal membrane potential Vm (given a reversal potential of Esyn) and the synaptic conductance gsyn. The term j represents the synaptic efficacy of each connection and noise is the background neuronal activity. The synaptic conductance gsyn (eqn (3)) was modelled with two alpha functions. This approach allowed us to independently control the rise and decay times of the conductance by adjusting two time constants: trise and tdecay. The conductance function starts at the time of the presynaptic spike j (when ), which is determined by the synaptic latency and latency jitter between the presynaptic and the postsynaptic neuron. Inhibitory synapses are included in the model by assigning negative values to j. The synaptic noise is modelled by creating random negative and positive deflections in the amplitude of Vm (the distribution of the noise-event amplitudes can be gaussian or uniform). Ideally, we would reproduce the noise of each neuron by using a sum of realistic EPSPs with different amplitudes and time courses. Unfortunately, we do not know enough about the characteristics of the multiple unitary EPSPs that make the background activity of the different neurons within the visual pathway. Consequently, we decided to use a more general form of random noise, which allows us to investigate the effect of modifying the amplitude distribution and time course of the noise events (Figs 6 and 7).
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Correlograms (N1 N2) obtained with different gaussian distributions of synaptic-event amplitudes, called here synaptic noise (average amplitude is constant; S.D. varies). The cartoon on the top-left illustrates the broadening of the distributions of noise amplitude. Increasing the S.D. of the noise made monosynaptic peaks slightly broader and in amplitude. Total number of N2 spikes: 10 000 for all three correlograms. On the left, examples of the EPSPs + synaptic noise used (shown as spike trigger averages; total time shown, 20 ms; EPSP amplitude, 3.6 mV). Symbols as in Fig. 3. Firing rates: N1, 75 spikes s–1; N2, 115–225 spikes s–1.
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Correlograms (N1 N2) obtained with different frequency spectrums of synaptic noise. The synaptic noise has been filtered by convolving it with exponential functions with different time constants (average noise and S.D. are constant; time constant varies). Modifying the frequency spectrum of the synaptic noise had a small influence on the width of the monosynaptic peak. Notice that, as the noise is smoothed, the correlograms start showing multiple peaks. These non-physiological correlogram shapes are also found when the noise is totally eliminated (see Fig. 12). On the left, examples of EPSPs + synaptic noise (shown as spike trigger averages; total time shown, 20 ms; EPSP amplitude, 3.6 mV). Symbols as in Fig. 3. Firing rates: N1, 100 spikes s–1; N2, 140 spikes s–1.
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With these three equations we define a new IAF model, based on synaptic currents, that includes some of the main physiological parameters that influence the time precision of action potentials and the shape of the monosynaptic crosscorrelogram. The model allowed us to manipulate 10 parameters independently for each neuron: firing rate, synaptic noise, EPSP rise time, EPSP duration at half-amplitude, EPSP amplitude, latency jitter, conduction delay, resting potential, threshold and refractory period. In most simulations only one parameter was modified while the others were kept constant at their default values. The default values were as follows: N1 firing rate: 25–100 spikes s–1; N1 resting potential: –70 mV; N1 threshold: –40 mV; N1 uniform noise: 2.1–3.6 nA; N2 resting potential: –70 mV; N2 threshold: –40 mV; N2 gaussian noise: μ = 2.6 nA, s = 0.5 nA; N2 EPSP rise time: 2.9 ms; N2 EPSP duration at half-amplitude: 8.7 ms; N2 EPSP amplitude (measured at the resting potential): 2.35 mV. The EPSP amplitude depends on the membrane potential. It is 0 for a membrane potential of 0 mV and maximum for a membrane potential of –80 mV. The values of EPSP rise time, duration and amplitude were all measured at the resting potential. The EPSP rise time was measured from the baseline to the peak of the EPSP. The EPSP duration was measured as the width of the EPSP at half-amplitude; N1 N2 weight (synaptic efficacy modelled as maximum conductance in eqn (2)): 8.75 nS for simulations using gaussian noise and 6.25 nS for simulations using uniform noise. The connection strength was defined by the weight. For a weight of 1.25 nS, the EPSP amplitude was 0.77 mV (measured at resting potential); N1 N2 synaptic delay: 1 ms; N1 N2 latency jitter: 0.5 ms.
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In some experiments we used three different default values to explore how pair combinations of parameters influence the shape of the monosynaptic peak in the correlogram. In these experiments we used the following default values: EPSP rise time: 0.5, 1.7 and 3.1 ms; EPSP duration: 2.4, 14.4 and 30.3 ms; threshold: –38.5, –37 mV (–64.75, –64.5 mV when the test parameter is EPSP rise time); synaptic noise: 0, 0.5 and 1 nA; EPSP/noise: 0.85, 2.47 and 5.26; latency jitter: 0, 0.8 and 1.5 ms.
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The parameter values explored in our simulations are inspired by physiological measurements of different connections across the early visual pathway. For example, according to Bloomfield & Sherman (1988), retinogeniculate EPSPs in the cat have mean amplitudes of 1.87 mV for X cells and 2.6 mV for Y cells, rise times of 0.61 ms for X cells and 0.43 ms for Y cells, total durations of 1.91 ms for X cells and 3.21 for Y cells and latency jitters of 0.35 ms for X cells and 0.17 for Y cells (see also Eysel, 1976). Stratford et al. (1996) measured cortical EPSPs in the cat that were thought to originate in thalamus and cortex. They measured the following values. Mean amplitudes: 1.9 mV (thalamic) and 0.2–1 mV (cortical); rise times: 1.1 ms (thalamic) and 1.1–1.4 ms (cortical); half-width duration (duration from 50% of peak amplitude in rising phase to 50% of peak amplitude in falling phase): 8.8 ms for thalamic and 10.7–12.7 ms for cortical; EPSP latency jitter: 0.3 ms for thalamic and 0.7 ms for cortical. All these values can vary within a wide range. For example, retinogeniculate EPSPs can be as large as 6 mV (Bloomfield & Sherman, 1988) in the cat. In the rat visual cortex, EPSP rise times can be longer than 3 ms and EPSP durations longer than 30 ms (Mason et al. 1991).
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Correlated firing
The correlated firing generated by the N1 N2 connection is revealed by cross-correlation analysis as a monosynaptic peak displaced from zero in the correlogram (Perkel et al. 1967; Moore et al. 1970; Gerstein & Perkel, 1972; Gerstein & Aertsen, 1985; Gerstein et al. 1985). In the correlograms illustrated in this paper, the X-axis shows the time delay between N1 and N2 paired spikes (bin size = 0.1 ms for all simulated correlograms and 0.5 ms for the correlograms from Fig. 2). Positive delays (right side of the correlogram) illustrate N1 spikes that preceded N2 spikes. Negative delays (left side of the correlogram) illustrate pairs of N2 spikes preceding N1 spikes.
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A–C, three different correlograms generated by retinogeniculate connections (A, from Alonso et al. 1996), geniculocortical connections (B, from Alonso et al. 2001) and corticocortical connections (C, from Alonso & Martinez, 1998) in the cat visual pathway (bin size, 0.5 ms). The retinogeniculate correlogram was obtained by simultaneously recording from s-potentials and geniculate spikes with two different electrodes (in a serendipitous recording from two geniculate neurons that received input from the same retinal afferent). The geniculocortical correlogram was obtained in simultaneous recordings from a geniculate cell and a layer 4 cortical simple cell (area 17). The corticocortical correlogram was obtained in simultaneous recordings from a layer 4 cortical simple cell and a complex cell in layers 2 + 3 (area 17). D, average width (and S.D.) of monosynaptic peaks generated by retinogeniculate, geniculocortical and corticocortical connections. Data were taken from Usrey et al. (1999) for retinogeniculate connections (simultaneous recordings from retinal cells and geniculate cells); from Alonso et al. (2001) for geniculocortical connections; and from Alonso & Martinez (1998) for connections from layer 4 to layers 2 + 3. We chose the 10 strongest monosynaptic peaks obtained from each of these studies (10 strongest retinogeniculate peaks; 10 strongest geniculocortical peaks; 10 strongest corticocortical asymmetric peaks). The peak width was measured at half-amplitude after subtracting the baseline (equivalent to measurements at 50% of peak height in the following figures).
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A monosynaptic peak was considered to be significant if it met two criteria. First, the left side of the correlogram (between –50 and –40 ms) contained more than 50 events. Second, the amplitude of the monosynaptic peak was four S.D.s above the baseline. The correlogram baseline (mean and S.D.) was calculated by averaging all correlogram bins between –50 and –40 ms. This interval (–50, –40 ms) was ideal for baseline measurements because all our simulated correlograms were flat with the exception of the region surrounding the peak. The peak height was calculated as follows. First, we calculated the maximum of the correlogram within the interval (–10, 10 ms) and averaged the 12 bins that surrounded the bin with the highest value. Then, the peak height was measured as the average maximum minus the baseline. The peak width was measured at 25 and 50% of the peak height by using the algorithm described in eqns (4)–(7) (non-significant peaks were assigned a peak width of 0).
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where vl is the number of spikes at a given level l of the peak height (25 or 50%), b is the baseline, C is the correlogram and p is the correlogram bin with the largest amplitude. mlpre is the bin where the summation of the correlogram to the left of the peak reaches a minimum and mlpos is the bin where the summation of the correlogram to the right of the peak reaches a maximum. The peak width (wl) was defined as the number of bins between mlpre and mlpos.
We measured how the peak width changed as we modified the value of a given parameter. For each parameter value (e.g. EPSP rise time = 2 ms), we simulated 100 correlograms and calculated the average. For example, if we used 40 different values of EPSP rise time, we obtained 40 different averages. These averages were then used to calculate a peak-width ratio: maximum average/minimum average (zeros were not included in the averages). Measurements of peak-width ratio were done only for parameter values that generated at least 50% of significant correlograms. We measured the peak width at 25 and 50% peak height. We used the same approach to measure changes in the peak rise time.
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Spikes were generated in the circuit by introducing uniform noise in the membrane potential of N1. Spike collection stopped when N2 generated 2000 spikes unless specified differently in the text (e.g. for the conditions in which N2 had very low firing rates we set a time limit to stop spike collection). In initial experiments we made multiple measurements stopping spike collection at 1000, 2000, 5000, 10 000 and 50 000 N2 spikes. A value of 2000 N2 spikes was chosen because it gave us reliable measurements while allowing us to run simulations within a reasonable amount of time. Similarly, we did simulations with different firing rates for N2 and chose a range that was a good compromise between simulation speed and measurement reliability.
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Results
Strong monosynaptic connections generate a precise correlated firing between the neuron input and the neuron target. The precision of this correlated firing can be measured by cross-correlation analysis as the width of the monosynaptic peak in the correlogram. Figure 1 (top) illustrates a typical monosynaptic peak generated by a strong direct connection between N1 and N2 (see Methods for model; see also Veredas et al. 2004). Positive bins show the number of times that neuron 1 (N1) fired before neuron 2 (N2), negative bins show the number of times that N1 fired after N2, and the zero bin shows the number of times that both neurons fired together (bin width = 0.1 ms). The narrow peak displaced from zero indicates that N1 tended to fire before N2 with a time course characteristic of a monosynaptic connection. The three grey lines are from bottom to top: the baseline of the correlogram, the 25% and 50% height levels used to measure peak width. The bottom of Fig. 1 shows simulated intracellular recordings from N1 (top) and N2 (bottom). N2 is shown hyperpolarized and with no synaptic noise to make the EPSP shapes generated by N1 spikes more visible.
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Top, correlogram generated by a monosynaptic connection between two simulated neurons (N1 and N2). The correlogram indicates the number of cases that N1 fired before N2 (positive bins), after N2 (negative bins) or simultaneously with N2 (zero bin). Bin size, 0.1 ms. The correlogram shows a characteristic monosynaptic peak with a fast rise time displaced from zero indicating that N2 tended to fire after N1. The dip preceding the monosynaptic peak (left side of zero) is characteristic of correlograms generated by strong monosynaptic connections. Its shape matches the shape of the autocorrelogram from the presynaptic cell (Moore et al. 1970; see also Fig. 2). In our stimulations, the width and amplitude of this dip could be modified by increasing the ratio of the EPSP/noise amplitude and/or by increasing the refractory period. The precision of the correlated firing generated by the monosynaptic connection was estimated by measuring the width of this monosynaptic peak. The three lines on the left show the baseline (bottom) and the two levels used to measure peak width (25 and 50% of peak height). Total number of N2 spikes: 50 000. Bottom, simulated intracellular recordings of N1 (top) and N2 (bottom). N2 has been hyperpolarized and no synaptic noise has been added to make more visible the shape of the EPSPs generated by N1.
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Monosynaptic peaks are usually identified in a correlogram by their narrow width and fast rise time (Perkel et al. 1967; Levick et al. 1972; Toyama et al. 1981; Tanaka, 1983; Mastronarde, 1987; Reid & Alonso, 1995; Swadlow, 1995; Alonso & Martinez, 1998; Usrey et al. 1999; Miller et al. 2001; Roy & Alloway, 2001). However, monosynaptic connections range widely in spike-timing precision (e.g. compare Alonso & Martinez, 1998; Usrey et al. 1999; Alonso et al. 2001; compare Bloomfield & Sherman, 1988; Mason et al. 1991). In fact, it is sometimes possible to determine whether a correlogram is generated by a retinogeniculate connection or a cortical connection (layer 4 layers 2 + 3) just based on the width of the monosynaptic peak. Figure 2A, B and C shows examples of the strongest monosynaptic peaks found in our physiological experiments at each level of the visual pathway: retina thalamus, thalamus cortical layer 4, layer 4 to layer 2 + 3 ((Alonso et al. 1996, 2001; Alonso & Martinez, 1998); the retinogeniculate correlogram was obtained from simultaneous recordings of s-potentials and geniculate spikes (S-potentials are large synaptic potentials, extracellularly recorded in geniculate cells, that are generated by single retinal afferents; see Usrey et al. 1999 for simultaneous recordings from retina and thalamus). The three correlograms share many features in common. All have narrow peaks superimposed on much slower correlations (shuffle correlograms for Fig. 2B and C are shown in Fig. 3 of Alonso et al. 1996 and Fig. 2 of Alonso & Martinez, 1998, respectively). All have peaks that are displaced from zero indicating that one cell tended to fire before the other. Finally, in all cases, the left side of the correlogram has a dip that matches the autocorrelogram of the presynaptic cell as would be expected from a strong monosynaptic connection (Moore et al. 1970).
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Top, three correlograms (N1 N2) obtained with different N2-EPSP rise times. As the rise time becomes longer the monosynaptic peaks become smaller and broader. The shapes of the EPSP and EPSP derivatives (EPSPd) used in the simulations are shown at the top of the correlograms. Note that the width of the monosynaptic peak in the correlogram resembles more closely the width of the EPSPd than the width of the EPSP. Total number of N2 spikes for all three correlograms, 5000. Bottom, plot showing an increase in peak width as the EPSP rise time increases. We ran 100 simulations for each value of EPSP rise time (100 correlograms) and represented the average and S.D. at 25% of peak height (black) and 50% of peak height (grey). A circuit representing the neurons involved is shown at the left: N2 receives excitatory input from N1 in addition to random synaptic noise (see Methods). Examples of the EPSPs used in the simulations are shown on top of the circuit diagram (shown without synaptic noise; total time shown, 10 ms; EPSP amplitude, 0.76 mV). Firing rates: N1, 107 spikes s–1; N2, 51–90 spikes s–1.
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There is also a noticeable difference among the three correlograms. The peak width is narrowest in the retinogeniculate connection (Fig. 2A) and widest in the corticocortical connection (Fig. 2C). This difference in peak width can be observed in individual cases (Fig. 2A, B and C) and on averages obtained from several cases (Fig. 2D; the retinogeniculate data for Fig. 2D was generously provided by M. Usrey, J Reppas and C. Reid from Usrey et al. 1999). A similar difference in peak width is found when comparing the thalamic synchrony generated by divergent retinal afferents with the cortical synchrony generated by divergent cortical afferents (e.g. compare Alonso et al. 1996; Alonso & Martinez, 1998).
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Computer simulations were used to identify the main factors that determine these differences in correlated firing precision. We used the simplest circuit possible – two neurons that are monosynaptically connected – and modified each parameter separately to study its contribution to correlated firing. Throughout the entire paper we measure the precision of the correlated firing as the width and rise time of the monosynaptic peak in the correlogram.
EPSP rise time
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In a classical paper, Fetz & Gustafsson (1983) showed that the precision of the correlated firing generated by a monosynaptic connection was strongly determined by the rise time of the EPSP and matched the EPSP derivative. EPSPs with fast rise times generated narrow monosynaptic peaks and EPSPs with slow rise times generated broader peaks (see also Matsumura et al. 1996). In contrast, the monosynaptic correlograms were found to resemble more closely the shape of the EPSP than the EPSP's derivative in Aplysia neurons (Moore et al. 1970) and the motoneurons of the crayfish claw (Lindsey & Gerstein, 1979).
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Measurements of EPSP rise time in experimental studies are usually obtained from averages and therefore they reflect the combination of two different variables: EPSP rise time and EPSP latency jitter (see below). Similarly, previous simulation studies (Surmeier & Weinberg, 1985) modified two parameters at the same time (EPSP rise time and EPSP amplitude) making it difficult to determine the contribution of the EPSP rise time in isolation. Our simulations allowed us to re-examine this important issue by generating a large number of correlograms using EPSPs that differed mainly in their rise times. We measured the width of the monosynaptic peak obtained with different values of EPSP rise time (Fig. 3). Examples of correlograms are shown at the top of the figure and the measurements for all correlograms at the bottom. Increasing the EPSP rise time had two main effects on the correlated firing generated by the connection N1 N2. First, it reduced the amplitude of the monosynaptic peak, and second, consistently with the findings of Fetz & Gustafsson (1983), it increased the width of the monosynaptic peak. Also consistent with Fetz & Gustafsson (1983), the width of the monosynaptic peak matched better the width of the EPSP derivative (EPSPd) than the width of the EPSP (illustrated at the top of the correlograms). We quantified the changes in peak width by calculating the average peak width for each value of rise time and then dividing the maximum by the minimum average (see Methods). The peak width ratio measured by this method was 5.53 (at 25% peak height) and 2.18 (at 50% peak height) and there was a strong correlation between EPSP rise time and peak width (0.74 and 0.91, respectively, Table 1).
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These results are in close agreement with Fetz & Gustafsson (1983; see also Knox, 1974) and extend previous results from Surmeier & Weinberg (1985) by showing that changes in EPSP rise time can modify the shape of the monosynaptic correlogram even if the EPSP duration remains relatively unchanged.
EPSP duration
It is widely believed that the EPSP duration has a small influence on the correlated firing generated by a monosynaptic connection (e.g. Kirkwood, 1979; Fetz & Gustafsson, 1983; Reid, 2001). This belief probably arises from previous experimental studies (Fetz & Gustafsson, 1983; Matsumura et al. 1996) and from the logical assumption that the EPSP decay is below threshold and should have a minor contribution to neuronal firing. Our results indicate quite the opposite. The EPSP duration may be one of the most important parameters in determining the correlated firing precision of some neurons. Figure 4 shows the results obtained by calculating multiple N1 N2 correlograms with different EPSP durations. EPSPs with short durations generated correlograms that resembled very closely those generated by retinogeniculate connections. In contrast, EPSPs with longer durations generated monosynaptic peaks with very long tails more typical of corticocortical connections. Long EPSPs hold the synaptic activity near threshold for long periods of time making the monosynaptic peaks broad. In the extreme case, if the EPSP is too long and the firing rate of N1 too high, N2 fires at its maximum capacity and the precise correlated firing between N1 and N2 fades away – the firing of N2 no longer depends on N1. This extreme situation is likely to be avoided in physiological circuits by disynaptic inhibition (Pouille & Scanziani, 2001; Wehr & Zador, 2003; Kuhn et al. 2004), sequences of inward–outward currents recruited by strong stimuli (Axmacher & Miles, 2004) or other mechanisms (e.g. synaptic depression). From all the parameters that we have studied, the EPSP duration was among the ones that generated the most pronounced changes in the width of the N1 N2 monosynaptic peak (width ratio at 25% was 11.87, Table 1). These results are consistent with data showing that action potential timing is more variable in neurons that generate long EPSPs than those generating short EPSPs (Fricker & Miles, 2000; see also Pouille & Scanziani, 2001; Wehr & Zador, 2003).
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Correlograms (N1 N2) obtained with different N2-EPSP durations. Increasing the EPSP duration made monosynaptic peaks broader. There was also an increase in the correlogram baseline due to the more effective temporal summation that led to a higher firing rate in N2. Notice the difference in the scale of the Y-axis (when EPSPs are brief most N2 spikes are precisely correlated with N1 spikes). As in Fig. 3, the shapes of the EPSP and EPSP derivatives (EPSPd) are shown at the top of the correlograms. Unlike in Fig. 3, the width of the monosynaptic peak in the correlogram resembles more closely the width of the EPSP than the width of the EPSPd. A cartoon of the circuit and examples of the EPSPs used are shown on the bottom left (shown without synaptic noise; total time shown, 80 ms; EPSP amplitude, 3.85 mV). Symbols as in Fig. 3. Firing rates: N1, 25 spikes s–1; N2, 15–175 spikes s–1.
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Threshold
Another parameter that could influence the precision of the correlated firing generated by a monosynaptic connection is the neuronal threshold. Neurons that have their baseline activity close to threshold fire more and could potentially generate broader correlograms because the EPSP is above threshold for a longer period of time (Kirkwood, 1979). As shown in Fig. 5, making the neuronal threshold higher (less negative) did not cause pronounced changes in peak width (see also Table 1). As expected, N2 became less excitable and there was a reduction in the number of events in the correlogram due to a reduction in the firing rate of N2. When the threshold was too high (e.g. –36 mV) N2 did not fire and the correlogram was non-significant (peak width = 0); when the threshold was too low (e.g. –41 mV) N2 fired at its maximum capacity totally independent from N1 (flat correlogram). Within the ranges that generated significant correlograms, changes in threshold had a relatively small influence on the width of the monosynaptic peak (Table 1). It should be noted, however, that these results were obtained using a static threshold and may differ from results obtained with thresholds that are more dynamic (Azouz & Gray, 2000, 2003).
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Correlograms (N1 N2) obtained with different N2 thresholds. Increasing the threshold generated only modest changes in the width of the monosynaptic peaks. The changes in the correlogram baseline are due to changes in the firing rate of N2. A cartoon of the circuit and examples of the EPSP used is shown on the bottom left (shown as spike trigger average; total time shown, 20 ms; EPSP amplitude, 2.2 mV). Firing rates: N1, 100 spikes s–1; N2, 165–35 spikes s–1.
Synaptic noise (amplitude and time course)
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The precision of the correlated firing generated by a monosynaptic connection (N1 N2) could be determined by the amplitude and time course of the multiple synaptic events that do not originate in N1. To test this possibility, we used a gaussian distribution of synaptic-event amplitudes (synaptic noise) and modified the S.D. of the gaussian while keeping the average current constant. For a S.D. of 0, the synaptic noise was a constant current and, as we increased the S.D., synaptic events of increasingly larger amplitude appeared in the baseline. Increasing the S.D. of the synaptic-noise distribution made monosynaptic peaks smaller because more N2 spikes were generated by synaptic events that did not originate in N1 (Fig. 6). In addition, the monosynaptic peaks became slightly wider (Table 1).
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In a different experiment, we modified the time course of the synaptic noise while keeping constant the average amplitude. In these simulations, the synaptic noise was convolved with exponential functions (e–t/) with distinct time constants () to generate noise with different frequency spectra (re-scaling the amplitude of the noise after filtering). These simulations were not able to generate large changes in the width of the monosynaptic peak (Fig. 7, Table 1; similar results were obtained by sampling the noise with different frequencies and reconstructing it with a cubic spline).
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Ratio of EPSP/synaptic noise
One of the most significant differences between retinogeniculate connections and corticocortical connections is in their synaptic strength. While a single retinal afferent can make more than one hundred synapses with a geniculate neuron (Hamos et al. 1987), the number of synapses per axon in geniculocortical and corticocortical connections is usually smaller than 10 (e.g. Freund et al. 1985; Feldmeyer et al. 2002). It is possible that differences in the precision of correlated firing between retinogeniculate connections and corticocortical connections are due to differences in the relative amplitude of the monosynaptic EPSPs. We investigated this possibility by modifying the relative amplitude of the postsynaptic EPSP with respect to the synaptic noise while keeping the mean firing rate of N2 constant.
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The results of these simulations indicate that the amplitude of the monosynaptic EPSP is strongly related with the amplitude of the monosynaptic peak in the correlogram but much less with its width or precision (see also Poliakov et al. 1997). Figure 8 shows simulated correlograms that were obtained with different EPSP/noise amplitude ratios. Increasing the ratio of the signal to noise had a pronounced effect on the amplitude of the monosynaptic peak but barely affected the peak width (Table 1).
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Correlograms obtained with three different ratios of EPSP/synaptic noise. For most EPSP/noise ratios, increments in the ratio did not generate pronounced changes in peak width. Only the smallest ratios (EPSP/noise < 1) generated slightly broader peaks. On the left, examples of EPSPs + synaptic noise used (shown as spike trigger averages; total time shown, 20 ms; Amplitude of EPSP at the bottom, 8 mV). Symbols as in Fig. 3. Firing rates: N1, 75 spikes s–1; N2, 110 spikes s–1.
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EPSP latency jitter
Monosynaptic connections also generate latency jitter, that is, the time delay between a presynaptic spike and the postsynaptic EPSP varies. The amount of temporal jitter depends on the type of neurons involved (Markram et al. 1997a; Feldmeyer & Sakmann, 2000; Feldmeyer et al. 2002). For example, in the developing rat neocortex, Markram et al. (1997a) measured fluctuations in EPSP latency of 1.5 ms between layer 5 neurons which is a very large value in comparison with the submillisecond jitter produced by retinogeniculate connections (Cleland et al. 1971a,b; Eysel, 1976; So & Shapley, 1979; Bloomfield & Sherman, 1988). The EPSP latency jitter reaches its highest values in polysynaptic connections and it is usually used as a criterion to distinguish between monosynaptic and disynaptic inputs (e.g. Ferster & Lindstrom, 1983). Therefore, to understand the role of polysynaptic inputs in generating precise correlated firing it is important to study how the polysynaptic correlated firing is affected by the total EPSP latency jitter (the total EPSP latency jitter depends on the jitter of each monosynaptic connection in the polysynaptic chain). Here, we studied the effect of increasing the latency jitter on the correlated firing generated by the connection N1 N2. Figure 9 (top) shows examples of correlograms obtained with three different jitter values. As we increased the latency jitter the monosynaptic peak became smaller and wider because many postsynaptic spikes were no longer precisely correlated with their presynaptic spikes (reduction in peak amplitude) and became distributed over longer periods of time (increase in the width of the peak tail). Changes in EPSP latency jitter also generated pronounced changes in the rise time of the monosynaptic peak (Table 1). As shown in Fig. 9, the connection N1 N2 frequently failed to generate precise correlated firing for jitter values larger than 1.6 ms. This result is important because it suggests that a polysynaptic connection has to generate very small latency jitter to generate correlograms that resemble monosynaptic connections. Typical polysynaptic correlogram peaks can be easily generated with large values of EPSP duration and latency jitter (Fig. 10, Table 1).
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Correlograms obtained with three different EPSP latency jitters (measured as the S.D. of a gaussian distribution for synaptic delay). Small increments in jitter produced a strong reduction in the amplitude of the monosynaptic peak and an increase in peak width. We could not generate significant correlograms when the latency jitter was larger than 1.6 ms. On the left cartoon of the circuit and examples of EPSPs used (total time shown, 40 ms; EPSP amplitude, 3.85 mV). Symbols as in Fig. 3. Firing rates: N1, 100 spikes s–1; N2, 150 spikes s–1.
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Correlograms obtained while increasing both the latency jitter and EPSP duration. As the latency jitter and the EPSP duration increases the monosynaptic peaks become weaker and broader. On the left, cartoon of the circuit and examples of EPSPs used (shown without synaptic noise; total time shown, 40 ms; EPSP amplitudes, 5.85 mV). Symbols as in Fig. 3. Firing rates: N1, 75 spikes s–1; N2, 44–330 spikes s–1.
Interactions between pairs of parameters
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The results presented above indicate that the width of a monosynaptic peak in a correlogram is mostly determined by the EPSP time course (rise time, duration and latency jitter). However, any conclusion from these results is limited by the fact that we did not explore all possible parameter combinations. In our simulations each parameter was modified independently while keeping the other parameters at default values. For example, to study how the width of the monosynaptic peak depended on EPSP rise time, we modified the value of EPSP rise time while keeping the EPSP duration at a default value of 8.7 ms. While this strategy allowed us to examine the contribution of each factor independently, the results might be different if we had used other default values. To address this limitation we repeated all the simulations described above using three different default values for each pair combination of parameters tested (see Methods for details on the default values used). For example, we studied how peak width changes with EPSP rise time (test parameter) and EPSP duration (default parameter) by running the simulations for EPSP rise time of Fig. 3 three times, each time using a different value of EPSP duration. The largest peak-width ratio obtained in the three simulations was plotted as a circle in an X–Y grid (Fig. 11A; the circle diameter represents the value of the ratio).
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Changes in peak width (A and C) and peak rise time (B and D) obtained with different pair combinations of parameters (measured at 25% of peak height). A, the circle diameter represents the value of the peak-width ratio. The circle with a number indicates the largest ratio measured within the entire grid (e.g. the largest peak-width ratio was 22). The peak-width ratio for each pair was obtained as follows. We changed the value of a specific parameter (test parameter) and obtained a peak-width ratio (maximum peak width/minimum peak width). We repeated the measurements three times using three different values of a specific default parameter (default parameter) and obtained three peak-width ratios. The maximum ratio obtained in the three measurements was selected and shown in this figure. For example, the circle with the number 22 (test parameter, EPSP rise time; default parameter, EPSP duration) was obtained by running simulations for EPSP rise time three times using EPSP duration of 2.4, 14.4 or 30.3 ms. The diagonal of the plot shows the results from Figs 3, 4, 5, 6, 8 and 9 obtained using only one default value (e.g. EPSP rise time/EPSP rise time, EPSPd = 8.7 ms). B, same as A, but measuring peak rise at 25% of peak height. C, the circle diameter represents the correlation index between peak width and the test parameter. The circle with a number (0.97) indicates the strongest correlation measured within the entire grid (e.g. the strongest correlation was obtained in the simulations for EPSP duration using an EPSP/noise ratio of 5.26). The correlation index for each pair of test parameter/default parameter was obtained as follows. We measured the correlogram peak widths obtained with multiple values of a test parameter (as in Figs 3–9) and obtained a correlation index between peak width and the test parameter. This measurement was repeated three times using three different values of a specified default parameter. The diagonal of the plot shows the results from Figs 3,4,5,6, 8 and 9 obtained using only one default value. D, same as C, but measuring peak rise at 25% of peak height. EPSPr, EPSP rise time; EPSPd, EPSP duration; Thres, threshold; Noise, synaptic noise; EPSP/n, ratio of EPSP/synaptic noise amplitudes; Jitter, EPSP latency jitter.
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The largest peak-width ratio within the entire grid (peak-width ratio = 22) was obtained in the simulations for EPSP rise time (test parameter) while using a EPSP duration of 30.3 ms. The simulations for EPSP duration and latency jitter also yielded large peak-width ratios. Figure 11B shows similar paired measurements for peak rise. The largest peak rise within the entire grid (peak-rise ratio = 16) was obtained by changing the latency jitter in the absence of synaptic noise (default value for noise = 0 nA). Finally, in most simulations the values of the test parameter were highly correlated with the values of peak width (Fig. 11C) and peak rise (Fig. 11D). The results from these paired simulations are consistent with our previous conclusion: the EPSP time course is the most important factor in determining the width of the monosynaptic peak.
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Simulations of monosynaptic correlograms in the absence of synaptic noise
The synaptic noise was a very important parameter in our simulations; without noise it was usually difficult to generate physiological correlograms. Figure 12 shows the results from a simulation in which we modulated the membrane potential of N2 with a ramp instead of noise (amplitude, 0.0025–0.005 nA; frequency, 33 Hz). In these simulations N2 generated bursts of spikes that were extremely precise in timing due to the lack of random fluctuations in the membrane potential. As a consequence of this artificially precise spike timing, the N1 N2 monosynaptic correlogram had multiple peaks that reproduced the burst pattern of N2. These multiple peaks are not found in physiological experiments probably because they are filtered by the synaptic noise. If gaussian noise is added to the simulations, the multiple peaks from the postsynaptic bursts fuse into a single peak. Consistently with physiological data, the monosynaptic peak is wider when the postsynaptic cell fire bursts instead of single spikes (Creutzfeldt et al. 1980; Gray et al. 1990; Eggermont et al. 1993). Figure 13 shows the results from simulations using a ramp to modulate the membrane potential of N2, but in this case N2 fired mostly single spikes (in these simulations N2 received monosynapic excitation and disynaptic inhibition from N1). In these simulations, the lack of bursts together with the disynaptic inhibition made the correlograms peak narrower than in the previous figure (the number of multiple peaks is reduced). Regardless of the differences in the shape of the correlograms, the results from the simulations using a ramp baseline were equivalent to the results using gaussian noise – increasing the EPSP rise time made monosynaptic peaks wider (notice that the measurements of peak width are mostly determined by the first peak of the correlogram; see Methods). Similar results were obtained by modulating the membrane potential of N2 with a sinusoid instead of a ramp.
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Correlograms obtained with different values of EPSP rise time (the membrane potential of N2 was modulated with a ramp instead of gaussian noise). Notice the multiple peaks in the correlogram due to the lack of variability in the membrane potential. Consistently with the results shown in Fig. 3, the monosynaptic peak closest to zero becomes weaker and slightly broader when the EPSP rise time increases. The amplitude of the ramp was 0.0025–0.005 nA and the frequency was 33 Hz. On the left, cartoon of the circuit and examples of EPSPs used (total time shown, 10 ms; EPSP amplitudes, 0.76 mV). Symbols as in Fig. 3. Firing rates: N1, 107 spikes s–1; N2, 128–153 spikes s–1.
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Correlograms obtained with different values of EPSP rise time (the membrane potential of N2 was modulated with a ramp instead of gaussian noise). N2 fired mostly single spikes and received both monosynaptic excitation and disynaptic inhibition from N1 (the disynaptic inhibition generates a dip on the right side of the correlogram). Consistently with the results from Fig. 3, the monosynaptic peak becomes weaker and broader as the EPSP rise time increases. On the left, cartoon of the circuit and examples of EPSPs used (total time shown, 10 ms; EPSP amplitudes, 0.76 mV). Symbols as in Fig. 3. Firing rates: N1, 107 spikes s–1; N2, 13–16 spikes s–1.
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While we were able to study the influence of EPSP rise time on the width of the monosynaptic peak using ramp/sinusoids as baseline, this approach did not work for EPSP duration. As the EPSP duration increased, temporal summation became so effective that N2 fired at its maximum capacity totally independent from N1. Without synaptic noise, increasing the EPSP duration on top of a ramp/sinusoid invariably led to a flat correlogram unless we modified several parameters at the same time (e.g. increase EPSP duration and reduce the firing rate of N1; increase EPSP duration and reduce the amplitude of the ramp). In other words, it was not possible to find a default parameter range that would work for all the EPSP durations when using a ramp/sinusoid instead of synaptic noise as the baseline. This problem could be partially solved by adding disynaptic inhibition to the circuit, but even then, the correlogram became flat for EPSP durations longer than 14 ms (Fig. 14).
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Correlograms obtained using different values of EPSP duration. In addition to monosynaptic excitation, N2 receives disynaptic inhibition from N1 and its membrane potential is modulated by a ramp instead of noise. Without synaptic noise, it was not possible to investigate the influence of EPSP duration on the shape of the monosynaptic peak. For EPSPs longer than 14 ms, the temporal summation was so effective that N2 fired at its maximum capacity, independently of N1. (If the parameters of the model were adjusted to reduce the temporal summation of the long EPSPs, shorter EPSPs were not able to make N2 fire). Therefore, when using a ramp instead of noise we cannot measure how changes in EPSP duration only, affect the shape of the monosynaptic correlogram. On the left, cartoon of the circuit and examples of EPSPs used (total time shown, 80 ms; EPSP amplitudes, 3.85 mV). Symbols as in Fig. 3. Firing rates: N1, 107 spikes s–1; N2, 2–254 spikes s–1.
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Discussion
The precision of the correlated firing generated by a monosynaptic connection varies across the visual pathway. It is highest in retinogeniculate connections and lower in geniculocortical and corticocortical (layer 4 to layers 2 + 3) connections. Here we have examined some of the factors that may be responsible for these differences in precision. Among all the parameters studied, the EPSP rise time, duration and latency jitter generated the most pronounced changes in the width of the monosynaptic peaks. This result is in close agreement with previous studies in cat motoneurons (Kirkwood & Sears, 1982; Fetz & Gustafsson, 1983; Gustafsson & McCrea, 1984; Cope et al. 1987; Poliakov et al. 1997), Aplysia neurons (Moore et al. 1970), the motoneurons of the crayfish claw (Lindsey & Gerstein, 1979) and more recent data from GABAergic neurons in neocortex (Galarreta & Hestrin, 2001). The result is also consistent with recent studies in the hippocampus showing that spike timing precision is greatly influenced by the shape of the EPSP, which can be prolonged by voltage-gated conductances or shorten by disynaptic inhibition (Fricker & Miles, 2000; Pouille & Scanziani, 2001; Axmacher & Miles, 2004).
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EPSPs generated by corticocortical connections tend to be longer in duration than the EPSPs generated by retinogeniculate connections and have more latency jitter (Eysel, 1976; Bloomfield & Sherman, 1988; Stratford et al. 1996; see Methods for detail). Therefore, the EPSP duration and latency jitter could be the main factors that determine the differences in correlated firing precision at different levels of the visual pathway. The differences in EPSP duration between thalamic and cortical neurons could be due to differences in the properties of the cell membrane (e.g. input resistance) and/or differences in the voltage-gated synaptic conductances that shape the EPSPs (Reyes & Fetz, 1993; Stuart & Sakmann, 1995; Andreasen & Lambert, 1999; Fricker & Miles, 2000). In the hippocampus, the EPSPs from pyramidal cells are greatly prolonged by the activation of a Na+ current; however, the EPSPs from inhibitory neurons show little voltage-dependent amplification and are shorter in duration (Fricker & Miles, 2000). The LGN and visual cortex could behave like inhibitory and pyramidal neurons in hippocampus; the EPSPs from LGN neurons might be less amplified by voltage-gated conductances than the EPSPs of cortical neurons. However, our model does not allow us to address this issue.
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Like with any model, ours is not free of limitations. By choosing an IAF model, we chose not to study important cellular properties that influence the shape of the EPSP. The EPSP is influenced by multiple factors that include membrane properties (e.g. time constant, input resistance), voltage-gated conductances, synaptic depression and disynaptic inhibition. The choice of the neuronal model was based on two criteria. First, we chose a model that would allow us to generate realistic monosynaptic correlograms (that resemble the physiological correlograms measured at the different stages of the visual pathway). Second, we chose a model that would allow us to generate thousands of correlograms in a reasonable amount of time (4000 correlograms per day). A crucial parameter in our model was the synaptic noise. In our hands, it was very difficult to simulate realistic monosynaptic peaks without using some type of synaptic noise (see Fig. 12). The gaussian synaptic noise simulates quite well the fluctuations observed in the membrane potential of intracellularly recorded neurons. In vitro intracellular recordings consistently show that monosynaptic EPSPs are embedded in gaussian noise (synaptic events with normally distributed amplitudes e.g. Stratford et al. 1996; Feldmeyer et al. 2005). The amplitude of this noise is clearly underestimated by the measurements in vitro since most of the connections are cut in this preparation; in vivo recordings show much noisier synaptic baselines (e.g. Hirsch et al. 1998).
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It should be emphasized that correlograms are shaped by other factors that were not explored here such as the stimulus conditions and changes in excitability (Perkel et al. 1967; Lee et al. 1977; Palm et al. 1988; Aertsen et al. 1989; Brody, 1998; Dorn & Ringach, 2003). However, it is important to notice that correlations generated by direct connections tend to be narrower than those generated simply by the stimulus (unless the stimulus is rapidly modulated and spatially diffuse; see Reid, 2001 for review). As shown here, most polysynaptic inputs should fail to generate precise correlated firing unless they generate very small values of EPSP latency jitter. In our simulations, monosynaptic peaks were very weak when the EPSP jitter was larger than 1.6 ms. Since most disynaptic connections generate jitter values larger than 2 ms (see below), the contribution of disynaptic inputs to the precise correlated firing generated by monosynaptic connections should be relatively small. Consistently, in cat visual cortex, the correlograms generated by disynaptically connected geniculate cells and layers 2+ 3 cells do not show significant monosynaptic peaks (Alonso & Martinez, 1998). Exceptionally, retinocortical disynaptic connections (retina LGN cortex) are able to generate precise correlated firing (Lee et al. 1977; Kara & Reid, 1999) probably due to the very small latency jitter of retinogeniculate connections (Cleland et al. 1971a, b; Eysel, 1976; So & Shapley, 1979; Mastronarde, 1987; Bloomfield & Sherman, 1988; Usrey et al. 1999).
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The argument that most polysynaptic inputs contribute little to the correlated firing generated by monosynaptic connections relies heavily on the values of latency jitter measured in physiological experiments. Recent studies using dual intracellular recordings combined with anatomical reconstructions are finding values of EPSP latency jitter of more than 1 ms in cortical neurons (e.g. Markram et al. 1997a; Feldmeyer & Sakmann, 2000; Feldmeyer et al. 2002). In addition, the jitter between the EPSP onset and the generation of a spike can be as large as 0.8 ms for geniculocortical connections in vivo which yields a total effective latency jitter of 1.8 ms for some monosynaptic connections in the cortex (Ferster & Lindstrom, 1983). Based on these estimates, a disynaptic connection should generate latency jitter values at least twice as large (3.6 ms). However, 3.6 ms is likely to be an underestimate; some data indicate that this value could be larger than 10 ms (Stuart & Sakmann, 1995).
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The EPSP duration varies among cortical neurons; consequently, not all corticocortical connections generate broad correlograms and not all corticocortical EPSPs have long durations. For example, inhibitory neurons in layer 4 of the somatosensory cortex can fire in very precise synchrony (Swadlow et al. 1998) probably due to strong geniculocortical connections (Swadlow, 1995; Swadlow et al. 2002) and electrical junctions (Galarreta & Hestrin, 1999; Gibson et al. 1999; Galarreta & Hestrin, 2001).
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It should be emphasized that our results are in good agreement with those from Fetz & Gustafsson (1983). Like Fetz & Gustafsson (1983), we find that the precision of the correlated firing generated by monosynaptic connections depends on the rise time of the EPSP: EPSPs with slow rise times generate broader monosynaptic peaks than EPSPs with fast rise times. Our results indicate, however, that the EPSP duration is also an important factor, a conclusion that is more in agreement with Moore et al. (1970), Lindsey & Gerstein (1979) and recent intracellular studies (Fricker & Miles, 2000; Pouille & Scanziani, 2001; Wehr & Zador, 2003; Axmacher & Miles, 2004). The differences in the results could be explained by differences in the EPSPs durations of the neurons studied. The width of the monosynaptic peaks studied by Fetz & Gustafsson (1983) ‘depended mostly on the EPSP rise time’ probably because the EPSPs of cat motoneurons are very short in duration. In contrast, the width of the monosynaptic peaks studied by Moore et al. (1970) depended mostly on the EPSP duration because the EPSPs recorded in Aplysia cells are very long (total duration > 200 ms as shown in Fig. 2 of Moore et al. 1970). It is important to remember that EPSP rise time and decay time are strongly correlated: EPSPs with slow rise times are usually longer in duration than EPSPs with fast rise times. Therefore, the results of Moore et al. (1970), Lindsey & Gerstein (1979) and Fetz & Gustafsson (1983) may not be so different after all.
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The finding that the correlated firing of monosynaptic connections could be strongly determined by the EPSP time course provides support for neural codes based on precise spike timing (Mainen & Sejnowski, 1995; Singer & Gray, 1995; Nowak et al. 1997; Reich et al. 1997; Diesmann et al. 1999; Reinagel & Reid, 2000). EPSPs with long durations may reduce spike-timing precision but they also increase the chances for two inputs to temporally interact. Therefore, the loss in synchrony precision from thalamus to visual cortex should not preclude the visual system to use spike timing for visual processing.
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