Dendritic Processing of Excitatory Synaptic Input in Hypothalamic Gonadotropin Releasing-Hormone Neurons
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《内分泌学杂志》
Emory University (C.B.R., K.J.S.), Atlanta, Georgia 30322
Mathematical Biosciences Institute (J.A.B.), Ohio State University, Columbus, Ohio 43210
Abstract
The activity of hypothalamic GnRH neurons results in the intermittent release of GnRH required for reproductive function. This intermittent neurosecretory activity has been proposed to reflect integration of intrinsic properties of and synaptic input to GnRH neurons. Determining the relative impact of synaptic inputs at different locations on the GnRH neuron is difficult, if not impossible, using only experimental approaches. Thus, we used electrophysiological recordings and neuronal reconstructions to generate computer models of GnRH neurons to examine the effects of synaptic inputs at varying distances from the soma along dendrites. The parameters of the models were adjusted to duplicate measured passive and active electrophysiology of cells from mouse brain slices. Our morphological findings reinforce the emerging picture of a complex dendritic structure of GnRH neurons. Furthermore, analysis of reduced morphology models indicated that this population of cells is unlikely to exhibit low-frequency tonic spiking in the absence of synaptic input. Finally, applying realistic patterns of synaptic input to modeled GnRH neurons indicates that synapses located more than about 30% of the average dendrite length from the soma cannot drive firing at frequencies consistent with neuropeptide release. Thus, processing of synaptic input to dendrites of GnRH neurons is probably more complex than simple summation.
Introduction
HYPOTHALAMIC GnRH neurons comprise the final common output of a central pattern generator whose intermittent release of GnRH is required for reproductive function. The intermittent release of GnRH has been proposed to reflect integration of intrinsic properties of and synaptic input to GnRH neurons (1).
Synaptic input to GnRH somata is sparse (2). Synaptic input to the dendrite, however, is greater than to somata (3, 4). Moreover, a recent study indicated that dendrites of GnRH neurons are much longer and morphologically more complex than previously appreciated (5). Finally, changes in synaptic input to dendrites of GnRH neurons occur between the breeding season and the anestrous season in sheep (6), suggesting that synaptic input to the dendrite may be an important regulator in GnRH release.
The contribution of synaptic input to the dendrite in control of GnRH neuron firing is unknown. On one hand, the dendrite of GnRH neurons in most species exhibits relatively little branching (3, 5, 6, 7). Branch points shunt synaptic input because ionic currents flow both toward the soma on the main dendrite and away from the soma at a branch point (8). Thus, the fairly unbranched dendrite of GnRH neurons minimizes shunting of current before it reaches the axon hillock (near the soma) where action potentials are generally initiated. On the other hand, the dendrite is thin and becomes progressively thinner distal to the soma (3, 5, 7), resulting in a small cross-sectional area. This dendritic morphology leads to high axial resistance and exponential decay of voltage induced by synaptic currents as a function of the length constant and the distance between the input and the soma. Thus, it is unclear whether synaptic inputs on dendrites could control firing due to passive decay of voltage changes before reaching the site of action potential initiation.
Determining the relative impacts of synaptic inputs at different locations on the GnRH neuron is difficult, if not impossible, using only experimental approaches. Therefore, we have constructed models of hypothalamic GnRH neurons based on electrophysiological and morphological properties to examine how synaptic input to dendrites contributes to firing.
Materials and Methods
Constructing model neurons
The electrical activity of neurons is the result of a complex interplay between ionic currents that flow through the cell membrane and along processes in the cells. These flows of current affect the distribution of charge on the cell membrane, and thus its voltage. The electrical properties of neurons can be grouped into two types; active and passive. The passive electrical properties include membrane capacitance, membrane leakage resistance, and axial resistance along processes. These passive properties are independent of voltage or time, and their effects will underlie all electrical activity of the cell. Active electrical properties can include voltage-gated ion channel conductances and time-dependant synaptic inputs.
The dependence of each of these electrical properties on voltage and time can be approximated with mathematical formulae, and these can be combined with the equations governing the dynamics of electrical circuits to generate models of the electrical activity of neurons. In the compartmental models used in this study, a given living neuron is modeled with a set of connected three-dimensional compartments that is derived based on the cell’s morphology. Each of these compartments has specified electrical properties. The computer model steps through time, calculating at each step the currents that will flow into and out of each compartment and the resulting changes in compartment voltage. In this way, the electrical activity of an entire living cell can be modeled, including the responses to simulated stimuli.
Tissue preparation
Hypothalamic slices in the hemisagittal orientation (300 μm) were prepared using a vibrating microtome (Slicer HR-2; Sigmann Elektronik, Hueffenhardt, Germany) from male GnRH-green fluorescent protein (GFP) mice in which GnRH neurons express GFP under the control of a 3.47-kb fragment of the mouse GnRH gene promoter (9). Our preliminary studies indicated that longer dendrites were present in the hemisagittal slice orientation than those which were first reported in coronal slices (10). The average age of mice (n = 15) was 77 ± 2.9 d, and the average weight was 24.2 ± 0.43 g. The Institutional Animal Care and Use Committee of Emory University approved all procedures.
Mice were anesthetized with halothane and decapitated. Brains were quickly removed and placed in cold (1–2 C), artificial cerebrospinal fluid (ACSF) solution containing (in mM): NaCl (125), NaHCO3 (24), KCl (2.5), CaCl2 (1), MgCl2 (1), and glucose (10), equilibrated with 95% O2/5% CO2, pH 7.3–7.4. Slices were incubated in ACSF for 1–2 h at 32 C, transferred to a recording chamber mounted on the stage of an upright microscope (Axioskop, Carl Zeiss Microimaging Inc., Thornwood, NY), and then continuously perifused with ACSF (32 C). GnRH neurons were identified through their GFP expression using epifluorescent excitation at 470 nm and a x60 water immersion objective. To prevent excessive exposure of the slices to the epifluorescent excitation, a shutter (Uniblitz, Vincent Associates, Rochester, NY) was used between the light source (AttoArc, Carl Zeiss) and the objective. Slices were illuminated 100 msec every 1.5 sec during identification.
Electrophysiology
Recordings were made with an Axoclamp 2B amplifier in bridge mode (Axon Instruments, Union City, CA). Pipettes (3–6 M) were fabricated from borosilicate glass (AM Systems, Carlsborg, WA) using a pipette puller (PP-83; Narishige, Tokyo, Japan) and coated with Sylgard 184 (Dow Corning, Midland, MI). Pipettes were filled with (in mM): K-gluconate (130), HEPES (10), EGTA (0.2), NaCl (6), MgCl2 (2), NaATP (4), NaGTP (0.4), spermine (0.05), glutathione (5), and biocytin (0.5%). Electrodes were positioned using piezoelectric micromanipulators (Luigs and Neumann, Ratingen, Germany). Pipette capacitance and series resistance were compensated for electronically.
Neurons were not specifically selected based on dendrite lengths. However, neurons with dendrites that were clearly cut (based on GFP expression) were not used. One neuron was recorded in each slice to ensure an accurate match between the electrophysiology and morphology. Because the morphology of the modeled cell was based on a reconstruction of the measured cell from the fixed slice, it was important to have only one biocytin-filled cell per slice, to avoid errors in cell identification.
Determining electrical properties of GnRH neurons
After establishing the whole cell recording configuration, a series of negative current injections were performed to measure the total input resistance of the cell. From a holding current of –0.05 nA (to inhibit spiking in the active state), the applied current was stepped for 1 sec each to –0.08, –0.11, –0.14, –0.17, and –0.2 nA, with the current being returned to the holding value for 1 sec in between each step. The input resistance was calculated from the ratio of the change in measured voltage to the change in applied current.
To create a computer model that accurately represents the electrical properties of living cells, and can thus be of predictive use, electrical activity under varying conditions should be measured. For the models created in this study, the soma and axon contained voltage-dependent conductances representing sodium, potassium, and calcium channels. To "tune" the active conductances of the model, data were taken of the response of cells in slices to various somatic current injections. Responses to injections of +0.5 nA for 0.5 msec (generally a single action potential), +0.03 nA for 50 ms (generally 1–3 spikes) and to +2 nA for 200 msec (generally a single spike followed by a sustained depolarization, followed by a recovery to resting membrane potential after the injection) were measured in a bath of ACSF as described above. After these active measurements, the passive electrical response of the cells was measured while bathing the slices in a solution designed to block voltage-gated conductances and synaptic inputs. Voltage-gated currents and synaptic conductances were blocked using the following solution as the bath perfusate (in mM): tetrodotoxin (TTX; voltage-gated sodium channel blocker; 0.001), 4-aminopyridine (4-AP; broad glutamate receptor antagonist; 2), tetraethylammonium (TEA; potassium channel blocker; 10), 6-cyano-7-nitroquinoxaline-2,3-dione [CNQX; (S)--amino-3-hydroxy-5-methyl-isoxazolepropionic acid (AMPA)-type glutamate receptor antagonist; 0.01], picrotoxin [-aminobutyric acid (GABA)-A receptor channel antagonist; 0.02], Cd2+ (calcium channel blocker; 0.2), Ni2+ (to broadly block synaptic transmission 2), Cs+ (to block the hyperpolarization activated cation current; 5).
The passive response of the cells was measured using the "short-cip" protocol described by Major et al. (11). This short current injection and the decay of the resultant transient voltage change are quite sensitive to subtleties of cell morphology and will allow more accurate determination of passive cable parameters by fitting of modeled and measured responses in a full morphology model (11, 12). Voltage responses were recorded in response to injection of 0.5 msec short current injection pulses (short-cips) of + 0.5 nA and –1 nA at the soma for 100 trials.
Biocytin detection for anatomical reconstruction
After recordings, slices were fixed overnight in 5% formaldehyde. Slices were then washed in potassium PBS) and incubated in 1% H2O2 with 2% albumin and 10% methanol in KPBS. Biocytin-containing neurons were visualized using the Vector ABC protocol (Vector Laboratories, Burlingame, CA). Slices were incubated overnight in 0.1% Triton-X, 2% BSA (both from Sigma Chemical, St. Louis, MO) and avidin-biotin-horseradish peroxidase complex (Vector Laboratories). Biocytin-labeled cells are visualized with 3,3' diaminobenzidine (0.06%, 1% NiCl, 0.003% H2O2, KPBS; pH 7.4) with nickel enhancement. Slices were treated with 0.1% osmium to optimally preserve morphology and then dehydrated in increasing concentrations of ethanol (30–100%), and coverslipped out of xylene in Permount (Sigma Chemical). Additional neurons with cut dendrites (based on biocytin visualization) were eliminated.
Generation of virtual GnRH neurons
Morphologies were determined using a three-dimensional reconstruction system (Neurolucida, MicroBright Field, Inc., Williston, VT). For each recovered GnRH neuron, a Neurolucida morphology file was generated by tracing the cell, and this morphology file was converted with the Cvapp program (www.compneuro.org) to a GENESIS morphology file (www.genesis-sim.org/GENESIS/).
The net result of this process is a file listing the Cartesian coordinates, lengths, orientations, and diameters of a series of connected cylindrical compartments, with each compartment corresponding to an actual section of the recovered biocytin-filled cell in the slice. In the GENESIS (GEneral NEural SImulation System) modeling environment, the voltages and currents in each compartment are calculated independently, and connected compartments communicate by exchanging messages much as currents flow between sections of living neurons. The dynamics of each compartment are defined by passive parameters such as membrane capacitance and resistance and axial resistance, and by equations describing the variable conductances of ion channels and synapses.
Dendrites and axons were distinguished based on diameter and trajectory. First, the axon of the GnRH neuron is thinner than the dendrite as it leaves the soma (7). Second, axons could be visualized projecting toward the base of the hypothalamus. It must be noted, however, that dendrites often turned after coursing various distances from somata and also projected toward the base of the brain (see Results). Given these findings, we used diameter of the projection as it leaves the soma as our main criterion for distinguishing axons and dendrites. Location of synapses was assigned to somata, proximal dendrites, or distal dendrites of GnRH neurons.
Tuning the passive model with the genetic algorithm
Data were gathered from 11 different cells. These data were analyzed for input resistance and an average short-cip response was generated for each cell. These average responses were computed from both the positive and negative cip and were scaled by the magnitude of the injected current. The resulting averaged, normalized cip response was fit with a three-exponential function, to extract three time constants (0, 1, and 2).
The response of a neuron to an injection of charge, from either an experimental current injection or a synapse, will consist of two main components. First, there will be a passive response, as the cell membrane changes polarization due to the change in charge, and current leaks out of the cell through voltage-independent leak channels, and charge is redistributed throughout the cell by axial conduction. The second, active part of the response is due to the activation and inactivation of voltage-dependent conductances. Thus, in building a model of a cell, it is important to first simulate the passive response of the cell as accurately as possible. Voltage responses of passive neurons to current injection pulses were simulated in GENESIS and were compared with the voltage responses of the corresponding living neuron. The simulated voltage response in each model GnRH neuron was matched to its electrophysiological data using an algorithm that simultaneously finds the best fit for the parameters membrane capacitance, Cm, membrane resistance, Rm, and axial resistance, Ra.
Due to the large parameter space of multicompartmental models, automated parameter optimization algorithms can be extremely useful for finding parameter sets producing desirable model activity. Vanier and Bower (13) demonstrated that genetic algorithms are among the most effective parameter-search methods for large parameter spaces (up to 23 parameters). In brief, the genetic algorithm works by creating an initial population of parameter sets with randomly chosen values (within defined bounds) for each parameter in a set. Each parameter set is rated by comparing the model activity with the parameter set against the defined activity in the living neuron. In an evolution-inspired fashion, new parameter sets are created by breeding the highest-rated parameter sets. The new offspring are then rated, and the best rated sets are bred to produce new offspring. The breeding continues throughout a set number of generations to converge upon a best parameter set. The quality of the fit between the modeled neuron and the same neuron during recordings was determined by comparing the negative root mean square of the simulated and real voltage responses. A "good" fit requires less than 1% error between the simulated voltage responses and the voltage response in the living neuron.
Creating the active model
The genetic algorithm was used to optimize passive parameters for all 11 measured and reconstructed cells. The two model cells with the best passive fits were used to build active models. These GnRH neurons were also representative of the two major morphological types we identified (see Results). Once the passive properties of each model were determined through optimization with the genetic algorithm, active voltage-gated channels, modeled with Hodgkin-Huxley equations, were added to soma and axon compartments. The kinetics of the channels were adapted from published models of GT1–7 cells (14, 15). The models included fast sodium (Na), delayed rectifier potassium (Kdr), and L-type calcium (CaL) conductances (14, 15, 16, 17).
For this conductance-based active model, cell membrane potential is calculated as:
(1)
is the passive membrane leakage current, and Iinj is injected current. Each of the ionic currents is described by an equation of the form: Ii = gi(Vm,t)(Vm – Ei), where the ionic conductance, gi, is a specified function of membrane potential and time, and the reversal potential, Ei, depends on the ionic species. The ionic conductances, in the formalism of Hodgkin and Huxley, were described in terms of a maximum conductance for each channel type (Na, (Kdr, and (CaL), and one or more gating variables. The gating variables model the activity of activation gates (m for Na and CaL and n for Kdr) and inactivation gates (h for Na). For this model, the three conductances were described by the equations:
(2)
(3)
(4)
The exponents on the gating variables, m, h, and n indicate that the Na conductance has three activation gates and one inactivation gate, and the K delayed rectifier and L-type Ca conductances have no inactivation gates, and 4 and 2 activation gates, respectively. The maximum conductance, i, represents the single channel conductance times the channel density. It is these conductance values that are tuned to make the model duplicate measured electrophysiological activity.
The kinetics of a particular type of channel are modeled by specifying the functional dependence of the gating variables on membrane voltage and time. The voltage dependence for each gating variable, x {m, h, n} is described by a steady-state value, x(V), and a time constant, x(V). The time course of the change in activation or inactivation for a change in potential from V1 to V2 is:
(5)
The voltage dependence of the steady-state gating variables and time constants are all modeled in the forms:
(6)
And
(7)
(8)
The values for the kinetic parameters s (= +1 or –1), ssx, dtx, (seconds), Vx, Kx, Vtx, and Ktx (mV) were all taken from the published values of (14, 15), with the exception of the fast sodium, where the half-activation potentials, Vm and Vh, were decreased by 14.6 mV, to more accurately duplicate spike threshold and amplitude. All of the time constants were divided by a rate factor of 5 to compensate for the fact that the data on which the models in (14, 15) were based were taken came from cells at 25 C, whereas the present study was based on brain slices measured at 32 C (see Ref.18 for an explanation of this temperature scaling of rate constants). Table 1 summarizes the values of the kinetic parameters used in this model.
The relative densities of these three types of channels (Na, (Kdr, and (CaL), were adjusted separately in the soma and axon, to duplicate measured active responses to current injection. The channel densities were tuned to simultaneously match measured resting membrane potential, and response to the three types of current injection described above. Each model neuron was tuned to duplicate the physiological responses of the corresponding living cell before being used to test simulated synaptic inputs.
Making synapses for simulated inputs to virtual GnRH neurons
Multiple synaptic phenotypes have been identified on GnRH neurons. We focused on the impact of excitatory inputs because whole cell recordings indicate GnRH neurons have low intrinsic rates of firing (9, 10), and therefore most observed activity will likely be the result of excitatory input. Glutamate is a major excitatory neurotransmitter in the hypothalamus (19) and has been implicated in control of GnRH secretion (20). Although GABA has also been reported to be excitatory in GnRH neurons (21), this remains controversial (22). Thus, we focused on glutamatergic inputs. Characteristics of glutamate currents in GnRH neurons have been well defined (9, 23, 24). The time course of AMPA inputs, which are similar in GnRH neurons to other neurons, was defined using a dual exponential function
(9)
with a 1 and 2 of 0.5 and 1.2 msec, respectively.
Bifurcation analysis
To aid in interpretation of model results, a bifurcation analysis of the model was performed. For this purpose, a single-compartmental version of the model was developed; this reduced model was then simulated using the software XPPAUT, developed by G. B. Ermentrout and available at ftp://ftp.math.pitt.edu/pub/bardware. A copy of the XPPAUT file containing the model is available from the authors upon request. The conductance densities in the soma of the reduced model were tuned to match active electrophysiological responses as described for the full model above.
Bifurcation analysis produces a diagram relating the voltage response of the model cell to applied current, thus providing a visualization of the threshold amount of injected current required to generate an action potential in the model. The response of a cell to injected current may vary depending on the state of the cell before current injection; all possible responses (such as remaining at rest or spiking) should appear in the bifurcation diagram.
Results
Morphological properties of GnRH neurons
Table 2 indicates the morphological characteristics of biocytin-filled GnRH neurons. As previously demonstrated, GnRH neurons typically have one main dendrite. However, 36% of dendrites branched, giving rise to secondary dendrites which could be relatively long [average: 786.4 (±169.8) μm]. Dendrites typically branched in more distal locations [414.4 (± 164.9) μm], but the location of branch points was highly variable. The main dendrite was on average 510.0 (±89.6) μm in length.
The anatomical locations (panels A and D), biocytin reconstructions (panels B and E) and GENESIS renderings (panels C and F) of two modeled GnRH neurons are shown in Fig. 1. Approximately half of GnRH neurons had dendrites that arched and sent projections back in the direction of somatic fields (see Fig. 1B). Most of these dendrites closely followed the trajectory of the GnRH axon. The branching cell in Fig. 1, B and C, was one of the two used in active modeling. Other GnRH neurons exhibited the morphology typically reported in this population of cell (Fig. 1, E and F). This second, bipolar cell was the other made into an active model for this study.
Figure 2, A and E, compares the fit of passive voltage response in living and modeled bipolar (A) and branching (E) GnRH neurons to short current injection pulses (Fig. 2, B and F). Figure 2, C and G, shows representative response of a living GnRH neuron (black trace) to a current injection pulse of +0.5 nA for 0.5 msec (Fig. 2, D and H) before the application of the pharmacological solution (to render the neuron passive) and the corresponding voltage response of the model (gray trace). In response to this current injection, living GnRH neurons exhibited one to two action potentials similar to published results in GnRH neurons (24, 25). Simulated neurons were tuned to exhibit similar responses. Both models were tuned to match the same measured response (from a third cell). Because the total area of the bipolar cell was less than that of the branching cell, the bipolar cell had less total capacitance. The total capacitance of the model bipolar cell was 29.75 pF, compared with 58.79 pF for the branching cell. This meant that a smaller current input was required to generate the same voltage response in the smaller cell (0.261 nA compared with 0.45 nA). Living and simulated neurons exhibited relatively low spontaneous rates of firing, consistent with findings in hypothalamic GnRH neurons in slices using whole cell recordings (9, 10).
The bifurcation diagram for the single-compartment model is shown in Fig. 3A. In the graph, injected current increases along the horizontal axis and membrane potential varies along the vertical axis. For a given amount of injected current, the diagrams show all mathematically possible responses; a response that may actually be observed is referred to as stable. Solid lines indicate stable resting membrane potentials. Filled circles mark the maximum and minimum voltages of spikes that may be observed. Dashed lines and open circles represent unstable membrane potentials or spikes, respectively. These unstable responses are of interest for the mathematical analysis but would not be observed experimentally or in numerical simulations of the model.
These diagrams show that the model cell would not fire in the absence of injected current. With no injected current, the model gives a resting membrane potential of –49.8 mV. With current injected at the soma, Fig. 3A shows that the resting membrane potential rises from –49.8 mV to –36.5 mV as the current is steadily increased from 0 to 0.019 nA. As the injected current increases beyond 0.0192 nA, the cell begins spiking. Injected currents greater than 0.408 nA result in sustained depolarization.
To understand the impact of relatively long axons and dendrites on firing, we compared responses to current injection in single and multicompartment models. It is difficult if not impossible to generate full bifurcation diagrams from compartmental models with hundreds of compartments. Therefore, we studied the single compartment model generated in XPP (from which the bifurcation diagram was derived), a single compartment model with identical parameters constructed in Genesis, and the multicompartment Genesis model of the branching cell (e.g. a model that includes the dendrite and axon). The single compartment XPP and Genesis models are nearly identical in their firing rates in response to current injection. In contrast, in the full morphology model the general shape of the response is similar but the spike rates are lower. Moreover, the range of injected currents that lead to spiking is narrower in the multicompartment model.
Single synaptic inputs and spike initiation
To assess the effect of attenuation of synaptic input with increasing distance from the soma, a single simulated AMPA synapse, with the mathematical form described above, was placed first in the soma compartment of each model, and then at progressively more distal compartments until the postsynaptic response was insufficient to initiate an action potential. For a single synapse at the soma, the minimum unitary conductance sufficient to initiate an action potential was 2360 pS for the bipolar cell, and 3170 pS for the branching cell. Once again, the smaller bipolar cell’s lower total capacitance made it more responsive to smaller synaptic conductance amplitudes (postsynaptic currents). This threshold conductance amplitude was insufficient to generate spikes from any distal compartments past than the soma (bipolar cell) or 22 μm from the soma (Fig. 4, A and D). With twice the threshold conductance amplitude, a single synaptic event was able to initiate action potentials at distances up to 257 μm from the soma on the bipolar cell, and for the branching cell, at up to 329 μm from the soma on the short dendrite branch, and 245 μm out on the longer branch (Fig. 4, B and E). With three times the threshold value (7080 pS for the bipolar cell and 9510 pS for the branching cell), a single input was effective up to 299 μm (bipolar cell) and 358 μm on the short branch, and 275 μm on the long branch of the branching cell (Fig. 4, C and F). These distances compare with total measured dendrite lengths of 381 μm for the bipolar cell and 1236 μm for the longest branch of the dendrite of the branching cell. Thus, even relatively large synaptic inputs are insufficient to drive spiking when applied at a majority of the distance along the total dendrite length.
Ongoing synaptic input
In living neurons, synaptic inputs are not single strong pulses but consist of multiple inputs from one or more presynaptic cells. To simulate this, five independent inputs, each firing at an average rate of 20 Hz, were placed at varying distances from the soma. The actual number of inputs is important only in the context of total input frequency because total input frequency is the number of synapses multiplied by input frequency of each synapse. Activation times for inputs were based on a Gaussian distribution of activation intervals. The maximum conductance of each of the randomly firing synapses was set at 1250 pS. This replicates a 70 pA postsynaptic current, some of the largest currents that are found in GnRH somata during bath application of glutamate (23). Figure 5 shows a representative response of the branching model GnRH neuron to identical synaptic trains applied at various distances from the soma. When synapses were placed on the soma, firing was induced at an average rate of 10.1 Hz (5A). This rate of firing is consistent with our findings using dynamic current clamping to apply simulated AMPA inputs to living GnRH neurons. Application of this conductance profile (5 inputs at 20 Hz each for a total input of 100 Hz with unitary conductance of 1250 pS/input) resulted in a firing rate of about 9 Hz (24). Thus, there is good agreement between the response to synaptic trains in our virtual GnRH neurons in the present study and living GnRH neurons in an earlier study (24). The rate of firing decreased with increasing distance from the inputs to the soma. The induced spike rate fell less than 2 Hz for inputs 328 μm (5B) from the soma on the short branch, and 245 μm (5C) from the soma on the long branch of the dendrite, respectively. The bipolar model cell showed similar response, although with higher induced spiking frequencies for more proximal compartments, due to the lower total capacitance.
Figure 6 summarizes the impact of dendrite length on firing rate as assessed by simulated somatic voltage. As discussed previously, the smaller bipolar cell exhibits a larger induced spike frequency for proximal inputs, but for compartments more distal than 125 μm, both models showed similar response. Thus, even a strong synaptic stimulus loses its ability to drive high frequency firing at a distance that is less than the average dendritic length.
Discussion
The present study is the first to construct multicompartmental models of hypothalamic GnRH neurons. Two earlier studies developed so-called point models of GT1–7 cells (14, 15). A point model differs from a multicompartmental model because it treats the neuron as occupying a single point location in space. Thus, the behavior is completely independent of neuronal morphology. In contrast, a multicompartment model incorporates morphological aspects of the neurons of interest.
To understand their fundamental behavior, we reduced the models to a single compartment and used geometric dynamical systems methods to systematically analyze the activity seen in the reduced model. We also used the multicompartmental models with full, defined morphology to understand how synaptic input to the dendrite of GnRH neurons would control firing.
Surprising new insights into the morphology of GnRH neurons
The morphology of GnRH neurons is generally described as simple bipolar with a vertical orientation in the hypothalamus. The recent efforts of Campbell et al. (5) using coronal slices have challenged this view regarding the morphology of GnRH neurons. In the present study, we used hemisagittal slices. Thus, we can provide additional, important insight to this emerging picture. First, as noted by Campbell et al. (5), half of primary dendrites exit a coronal slice, thereby preventing measurement of their total length. In our hemisagittal slice orientation, the average length was longer than the average length reported in coronal slices (500 vs. 300 μm). Moreover, three of our filled dendrites had lengths that exceeded 1200 μm in length, somewhat longer than the maximum lengths reported by Campbell et al. (5). However, even in our slice orientation dendrites frequently extended beyond the plane of the slice. Thus, both studies are in agreement that dendrites of GnRH neurons are much longer than previously appreciated and likely extend into distal hypothalamic and perhaps extrahypothalamic regions.
Our findings support the observations that GnRH dendrites change directions after leaving the cell body (5), and we can extend this observation by demonstrating the trajectory of these dendrites. Using the hemisagittal slice orientation, our findings provide evidence that dendrites of about half of GnRH neurons closely follow the path of the axon. These dendrites may project to the median eminence. Several neurotransmitters have been suggested to regulate GnRH secretion at the level of the median eminence by acting on axons or nerve terminals. Synapses, however, are not generally identified on these structures. Our findings suggest that regulation of GnRH secretion at the level of the median eminence could occur through synapses on the dendrites of GnRH neurons.
Understanding the firing behavior in the reduced model
Bifurcation analysis of the reduced model predicts that sustained firing occurs only in the presence of continuously injected current of sufficient strength. This finding suggests that the mechanisms leading to repetitive firing in GnRH neurons may differ from those in other neurosecretory neurons.
Based on bifurcation analysis, sustained firing in GnRH neurons appears to depend on currents arising from extrinsic sources (i.e. synaptic inputs). Extrinsic current destabilizes the resting membrane potential and the GnRH neuron fires action potentials. These findings are consistent with dynamic-clamping experiments that indicate on-going synaptic input is required to maintain firing in GnRH neurons (24). In contrast, cultured GnRH neurons and GT1–7 cells exhibit spontaneous firing and GnRH secretion (1, 25). The spontaneous release of GnRH pulses in culture has formed the basis for the theory that intrinsic cellular properties can account for intermittent firing and pulsatile GnRH release. However, intrinsic properties can be significantly altered in culture. In GT1–7 cells, intervals between GnRH pulses and Ca2+ oscillations decrease from 80–21 min after 30 d in culture (25). Culturing has been shown to induce up-regulation of ion channels and alterations in firing properties such that firing patterns shift from irregular to burst firing despite the absence of bursting behavior in the intact ganglion (26, 27). Moreover, two recent studies indicate expression of calcium channels in hypothalamic GnRH neurons in short-term culture is dramatically lower and differs in phenotype from calcium channels expressed in GT1–7 cells and primary GnRH neurons in long-term culture (16, 17). Therefore, based on the present findings, it appears that the cellular mechanisms accounting for pulses from GT1–7 cells and cultured neurons may not be the same as those in hypothalamic GnRH neurons.
The above consideration notwithstanding, the model of LeBeau et al. (15) faithfully reproduces the firing patterns of GT1–7 cells. The GT1–7 cell model of LeBeau et al. (15) had several currents in addition to the ones used in the present models. Other currents in the GT1–7 model, in particular the store-operated Ca2+ channel (ISOC) and the nonselective inward cation current (Id), would be expected to have an effect on intrinsic spiking, bursting, and their inclusion in simulations of GT1–7 cells resulted in models that replicated experimental findings. Moreover, these conductances may be important in several modulatory responses. For our study of the mechanisms of spike generation and the effects of synaptic input, only the three strongest currents in the GT1–7 model were used. Additional currents were not necessary to duplicate the measured electrophysiology in the cells studied here, and were therefore not included. It should be emphasized, however, that the activity of the GT1–7 cells as described by LeBeau et al. (15) is different from that of the cells measured in this study, and thus the intrinsic currents may be expected to differ as well. In this regard, GT1–7 cells are derived from a clonal cell line, and their behavior (and conductances) may represent those of a specific subpopulation of GnRH neurons in the hypothalamus. Such differences in expression of ionic conductances would account for functional heterogeneity in hypothalamic GnRH neurons (28) and underscores the need for extensive study of conductance phenotypes expressed in hypothalamic GnRH neurons.
Understanding the impact of synaptic inputs in the multicompartmental model
The long dendrites of GnRH neurons potentially provide an extensive substrate for synaptic integration. Classical cable theory, however, predicts attenuation of signals in dendrites due to membrane capacitance and leak resistance (29, 30, 31, 32). In theory, attenuation of fast synaptic currents would become near-total in extended thin dendrites, like those of GnRH neurons. Experimental approaches such as local electrical stimulation to activate putative inputs to dendrite would provide only ambiguous results. Experimentally, one can stimulate fibers in the region of a dendrite, but the actual location of the presynaptic nerve terminal on the GnRH neuron cannot be determined. Thus, in the present study, we used experimentally based models of GnRH neurons to directly test the impact of synaptic input to specific regions of the dendrite. We determined where, on the extended dendrite, synapses could reside and still drive the repetitive firing that appears to be required for neuropeptide secretion (33). The pattern and frequency of firing that releases GnRH is unknown, but vasopressin neurons fire phasically with mean rates during bursts of 5–15 Hz during periods of hormone secretion (33). Moreover, at least 2 Hz is required to release enough GnRH to evoke postsynaptic events (34). During release of LH, single units extracted from multiunit activity volleys in the primate hypothalamus exhibit firing rates of 2–20 Hz (35). Thus, at least 2 Hz firing is likely required to drive GnRH release. In hypothalamic slices (9, 10), dispersed somata of GnRH neurons (23) and in cultures of GnRH neurons from the1) firing frequencies of at least 2 Hz have been defined. Therefore, we looked for dendritic locations where activation of synapses would result in firing frequencies of at least 2 Hz.
We focused on determining the impact of AMPA-type glutamatergic inputs because AMPA-type receptors appear to be the dominant form of glutamatergic receptors in GnRH neurons (9). Moreover, AMPA-type ligands are more effective than NMDA ligands at inducing repetitive firing in pharmacological experiments on GnRH somata (23). Other synaptic phenotypes have been identified on GnRH neurons (4, 36), some of which may provide additional excitation. For example, GABAergic inputs have been suggested to be excitatory in GnRH neurons (21). The reversal potential for GABA excitatory inputs in (21) was –36.5 mV. In contrast, the reversal potential for AMPA inputs is 0 mV. Thus, the electrochemical driving force (Vm–Erev) is larger for AMPA-mediated currents than the driving force on GABAergic currents by 36.5 mV. The impact of these other synaptic phenotypes remains to be determined. However, examining the impact of AMPA-type inputs provides the best case scenario for independent control of firing by excitation for the two major fast neurotransmitter phenotypes in the hypothalamus (e.g. glutamate and GABA).
We found in simulations of morphologically reconstructed GnRH neurons that AMPA-type excitatory post-synaptic currents (EPSC) and resulting excitatory post-synaptic potentials (EPSP) were significantly attenuated over a substantial portion of the dendrite length. Accordingly, even relatively high frequency, large amplitude currents derived from synaptic input could not drive repetitive firing when placed at dendritic locations more distal than less than half the actual observed dendrite length. The finding that synaptic input to a majority of the dendrite has little to no effect on somatic voltage raises significant issues for synaptic processing in GnRH neurons. The number of synapses on GnRH somata is limited. In male rats, there are two to 12 synapses on the somata of GnRH neurons when examined at the electron microscopic level using serial thin sections (2). Earliest reports using randomly selected sections for quantification suggested that most GnRH neurons had either no synapses (37) or less than one synapse/cell (3). The more recent analysis using serial sections, however, revealed that all GnRH neurons have at least two synapses, with an average of about seven synapses on somata (2). Thus, at the level of the somata, GnRH neurons are sparsely innervated suggesting that dendrites may be the principle sites of synaptic regulation. In mice, synaptic inputs to GnRH neurons appear to be qualitatively similar to other models (4, 38) with approximately 30% higher synaptic density on the proximal dendrite relative to the cell body. Moreover, synaptic density on dendrites increases during the breeding season in sheep when GnRH secretion is elevated (6). However, the present study indicates that only synapses located on somata and very proximal dendrites are capable of controlling somatic voltage. Thus, the findings of the present study indicate that, although the dendrite may receive extensive synaptic input, the morphology of the dendrite limits synaptic efficacy in the context of control of firing.
The finding that distal dendritic synapses do not alter firing calls into question the physiological significance of changes in input to the dendrite. It should be noted, however, that our model makes the assumption that the dendrite of the GnRH neuron is passive. This assumption is consistent with the dominant perspective in neuronal signaling, namely that action potentials are initiated in the soma/axon hillock region where voltage-sensitive sodium channels are clustered. This classical view of neuronal processing considers dendrites largely as passive conduits that relay changes in voltage from synapses to the axon hillock. In some populations of neurons, however, dendrites express active conductances, and in particular voltage-gated sodium channels. This endows dendrites with the ability to exhibit action potentials generally through back propagation of action potentials initiated at somata (39, 40, 41 ; see Ref.42 for review). Recent studies have indicated that in some circumstances, action potentials can be initiated in the dendrite through activation of voltage-sensitive sodium channels by excitatory inputs (43, 44, 45, 46, 47). If this were the case, the dendrite could be a potential site of synaptic integration. Additionally, dendrites with active properties could underlie synchronized firing in GnRH neurons via a common excitatory synaptic input and, as suggested by others, direct electrical communication between dendrites (48). It is currently unknown whether the dendrites of GnRH neurons possess active properties. This consideration notwithstanding, the findings of the present study indicate that purely passive integration is unlikely to be a suitable mechanism for the processing of synaptic input over much of the dendrite length.
Acknowledgments
We thank Dieter Jaeger for permitting the use of his laboratory for the electrophysiological components of this work. We thank Dr. Arthur Sherman for providing his ordinary differential equation files from 14 and 15 to K.J.S. during her participation in Methods in Computational Neuroscience at the Marine Biological Laboratory at Woods Hole.
Footnotes
This work was supported by HD-045436 (to K.J.S.) Preliminary data collection in coronal slices was supported by NS-010355. Support for the course at Woods Hole was provided by MH-062204.
A preliminary report of these data were presented in abstract form at the Annual Meeting of The Endocrine Society Meeting in 2004.
First Published Online December 22, 2005
Abbreviations: ACSF, Artificial cerebrospinal fluid; AMPA, (S)--amino-3-hydroxy-5-methyl-isoxazolepropionic acid; CaL, L-type calcium; GABA, -aminobutyric acid; GFP, green fluorescent protein; Kdr, delayed rectifier potassium.
Accepted for publication December 9, 2005.
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Mathematical Biosciences Institute (J.A.B.), Ohio State University, Columbus, Ohio 43210
Abstract
The activity of hypothalamic GnRH neurons results in the intermittent release of GnRH required for reproductive function. This intermittent neurosecretory activity has been proposed to reflect integration of intrinsic properties of and synaptic input to GnRH neurons. Determining the relative impact of synaptic inputs at different locations on the GnRH neuron is difficult, if not impossible, using only experimental approaches. Thus, we used electrophysiological recordings and neuronal reconstructions to generate computer models of GnRH neurons to examine the effects of synaptic inputs at varying distances from the soma along dendrites. The parameters of the models were adjusted to duplicate measured passive and active electrophysiology of cells from mouse brain slices. Our morphological findings reinforce the emerging picture of a complex dendritic structure of GnRH neurons. Furthermore, analysis of reduced morphology models indicated that this population of cells is unlikely to exhibit low-frequency tonic spiking in the absence of synaptic input. Finally, applying realistic patterns of synaptic input to modeled GnRH neurons indicates that synapses located more than about 30% of the average dendrite length from the soma cannot drive firing at frequencies consistent with neuropeptide release. Thus, processing of synaptic input to dendrites of GnRH neurons is probably more complex than simple summation.
Introduction
HYPOTHALAMIC GnRH neurons comprise the final common output of a central pattern generator whose intermittent release of GnRH is required for reproductive function. The intermittent release of GnRH has been proposed to reflect integration of intrinsic properties of and synaptic input to GnRH neurons (1).
Synaptic input to GnRH somata is sparse (2). Synaptic input to the dendrite, however, is greater than to somata (3, 4). Moreover, a recent study indicated that dendrites of GnRH neurons are much longer and morphologically more complex than previously appreciated (5). Finally, changes in synaptic input to dendrites of GnRH neurons occur between the breeding season and the anestrous season in sheep (6), suggesting that synaptic input to the dendrite may be an important regulator in GnRH release.
The contribution of synaptic input to the dendrite in control of GnRH neuron firing is unknown. On one hand, the dendrite of GnRH neurons in most species exhibits relatively little branching (3, 5, 6, 7). Branch points shunt synaptic input because ionic currents flow both toward the soma on the main dendrite and away from the soma at a branch point (8). Thus, the fairly unbranched dendrite of GnRH neurons minimizes shunting of current before it reaches the axon hillock (near the soma) where action potentials are generally initiated. On the other hand, the dendrite is thin and becomes progressively thinner distal to the soma (3, 5, 7), resulting in a small cross-sectional area. This dendritic morphology leads to high axial resistance and exponential decay of voltage induced by synaptic currents as a function of the length constant and the distance between the input and the soma. Thus, it is unclear whether synaptic inputs on dendrites could control firing due to passive decay of voltage changes before reaching the site of action potential initiation.
Determining the relative impacts of synaptic inputs at different locations on the GnRH neuron is difficult, if not impossible, using only experimental approaches. Therefore, we have constructed models of hypothalamic GnRH neurons based on electrophysiological and morphological properties to examine how synaptic input to dendrites contributes to firing.
Materials and Methods
Constructing model neurons
The electrical activity of neurons is the result of a complex interplay between ionic currents that flow through the cell membrane and along processes in the cells. These flows of current affect the distribution of charge on the cell membrane, and thus its voltage. The electrical properties of neurons can be grouped into two types; active and passive. The passive electrical properties include membrane capacitance, membrane leakage resistance, and axial resistance along processes. These passive properties are independent of voltage or time, and their effects will underlie all electrical activity of the cell. Active electrical properties can include voltage-gated ion channel conductances and time-dependant synaptic inputs.
The dependence of each of these electrical properties on voltage and time can be approximated with mathematical formulae, and these can be combined with the equations governing the dynamics of electrical circuits to generate models of the electrical activity of neurons. In the compartmental models used in this study, a given living neuron is modeled with a set of connected three-dimensional compartments that is derived based on the cell’s morphology. Each of these compartments has specified electrical properties. The computer model steps through time, calculating at each step the currents that will flow into and out of each compartment and the resulting changes in compartment voltage. In this way, the electrical activity of an entire living cell can be modeled, including the responses to simulated stimuli.
Tissue preparation
Hypothalamic slices in the hemisagittal orientation (300 μm) were prepared using a vibrating microtome (Slicer HR-2; Sigmann Elektronik, Hueffenhardt, Germany) from male GnRH-green fluorescent protein (GFP) mice in which GnRH neurons express GFP under the control of a 3.47-kb fragment of the mouse GnRH gene promoter (9). Our preliminary studies indicated that longer dendrites were present in the hemisagittal slice orientation than those which were first reported in coronal slices (10). The average age of mice (n = 15) was 77 ± 2.9 d, and the average weight was 24.2 ± 0.43 g. The Institutional Animal Care and Use Committee of Emory University approved all procedures.
Mice were anesthetized with halothane and decapitated. Brains were quickly removed and placed in cold (1–2 C), artificial cerebrospinal fluid (ACSF) solution containing (in mM): NaCl (125), NaHCO3 (24), KCl (2.5), CaCl2 (1), MgCl2 (1), and glucose (10), equilibrated with 95% O2/5% CO2, pH 7.3–7.4. Slices were incubated in ACSF for 1–2 h at 32 C, transferred to a recording chamber mounted on the stage of an upright microscope (Axioskop, Carl Zeiss Microimaging Inc., Thornwood, NY), and then continuously perifused with ACSF (32 C). GnRH neurons were identified through their GFP expression using epifluorescent excitation at 470 nm and a x60 water immersion objective. To prevent excessive exposure of the slices to the epifluorescent excitation, a shutter (Uniblitz, Vincent Associates, Rochester, NY) was used between the light source (AttoArc, Carl Zeiss) and the objective. Slices were illuminated 100 msec every 1.5 sec during identification.
Electrophysiology
Recordings were made with an Axoclamp 2B amplifier in bridge mode (Axon Instruments, Union City, CA). Pipettes (3–6 M) were fabricated from borosilicate glass (AM Systems, Carlsborg, WA) using a pipette puller (PP-83; Narishige, Tokyo, Japan) and coated with Sylgard 184 (Dow Corning, Midland, MI). Pipettes were filled with (in mM): K-gluconate (130), HEPES (10), EGTA (0.2), NaCl (6), MgCl2 (2), NaATP (4), NaGTP (0.4), spermine (0.05), glutathione (5), and biocytin (0.5%). Electrodes were positioned using piezoelectric micromanipulators (Luigs and Neumann, Ratingen, Germany). Pipette capacitance and series resistance were compensated for electronically.
Neurons were not specifically selected based on dendrite lengths. However, neurons with dendrites that were clearly cut (based on GFP expression) were not used. One neuron was recorded in each slice to ensure an accurate match between the electrophysiology and morphology. Because the morphology of the modeled cell was based on a reconstruction of the measured cell from the fixed slice, it was important to have only one biocytin-filled cell per slice, to avoid errors in cell identification.
Determining electrical properties of GnRH neurons
After establishing the whole cell recording configuration, a series of negative current injections were performed to measure the total input resistance of the cell. From a holding current of –0.05 nA (to inhibit spiking in the active state), the applied current was stepped for 1 sec each to –0.08, –0.11, –0.14, –0.17, and –0.2 nA, with the current being returned to the holding value for 1 sec in between each step. The input resistance was calculated from the ratio of the change in measured voltage to the change in applied current.
To create a computer model that accurately represents the electrical properties of living cells, and can thus be of predictive use, electrical activity under varying conditions should be measured. For the models created in this study, the soma and axon contained voltage-dependent conductances representing sodium, potassium, and calcium channels. To "tune" the active conductances of the model, data were taken of the response of cells in slices to various somatic current injections. Responses to injections of +0.5 nA for 0.5 msec (generally a single action potential), +0.03 nA for 50 ms (generally 1–3 spikes) and to +2 nA for 200 msec (generally a single spike followed by a sustained depolarization, followed by a recovery to resting membrane potential after the injection) were measured in a bath of ACSF as described above. After these active measurements, the passive electrical response of the cells was measured while bathing the slices in a solution designed to block voltage-gated conductances and synaptic inputs. Voltage-gated currents and synaptic conductances were blocked using the following solution as the bath perfusate (in mM): tetrodotoxin (TTX; voltage-gated sodium channel blocker; 0.001), 4-aminopyridine (4-AP; broad glutamate receptor antagonist; 2), tetraethylammonium (TEA; potassium channel blocker; 10), 6-cyano-7-nitroquinoxaline-2,3-dione [CNQX; (S)--amino-3-hydroxy-5-methyl-isoxazolepropionic acid (AMPA)-type glutamate receptor antagonist; 0.01], picrotoxin [-aminobutyric acid (GABA)-A receptor channel antagonist; 0.02], Cd2+ (calcium channel blocker; 0.2), Ni2+ (to broadly block synaptic transmission 2), Cs+ (to block the hyperpolarization activated cation current; 5).
The passive response of the cells was measured using the "short-cip" protocol described by Major et al. (11). This short current injection and the decay of the resultant transient voltage change are quite sensitive to subtleties of cell morphology and will allow more accurate determination of passive cable parameters by fitting of modeled and measured responses in a full morphology model (11, 12). Voltage responses were recorded in response to injection of 0.5 msec short current injection pulses (short-cips) of + 0.5 nA and –1 nA at the soma for 100 trials.
Biocytin detection for anatomical reconstruction
After recordings, slices were fixed overnight in 5% formaldehyde. Slices were then washed in potassium PBS) and incubated in 1% H2O2 with 2% albumin and 10% methanol in KPBS. Biocytin-containing neurons were visualized using the Vector ABC protocol (Vector Laboratories, Burlingame, CA). Slices were incubated overnight in 0.1% Triton-X, 2% BSA (both from Sigma Chemical, St. Louis, MO) and avidin-biotin-horseradish peroxidase complex (Vector Laboratories). Biocytin-labeled cells are visualized with 3,3' diaminobenzidine (0.06%, 1% NiCl, 0.003% H2O2, KPBS; pH 7.4) with nickel enhancement. Slices were treated with 0.1% osmium to optimally preserve morphology and then dehydrated in increasing concentrations of ethanol (30–100%), and coverslipped out of xylene in Permount (Sigma Chemical). Additional neurons with cut dendrites (based on biocytin visualization) were eliminated.
Generation of virtual GnRH neurons
Morphologies were determined using a three-dimensional reconstruction system (Neurolucida, MicroBright Field, Inc., Williston, VT). For each recovered GnRH neuron, a Neurolucida morphology file was generated by tracing the cell, and this morphology file was converted with the Cvapp program (www.compneuro.org) to a GENESIS morphology file (www.genesis-sim.org/GENESIS/).
The net result of this process is a file listing the Cartesian coordinates, lengths, orientations, and diameters of a series of connected cylindrical compartments, with each compartment corresponding to an actual section of the recovered biocytin-filled cell in the slice. In the GENESIS (GEneral NEural SImulation System) modeling environment, the voltages and currents in each compartment are calculated independently, and connected compartments communicate by exchanging messages much as currents flow between sections of living neurons. The dynamics of each compartment are defined by passive parameters such as membrane capacitance and resistance and axial resistance, and by equations describing the variable conductances of ion channels and synapses.
Dendrites and axons were distinguished based on diameter and trajectory. First, the axon of the GnRH neuron is thinner than the dendrite as it leaves the soma (7). Second, axons could be visualized projecting toward the base of the hypothalamus. It must be noted, however, that dendrites often turned after coursing various distances from somata and also projected toward the base of the brain (see Results). Given these findings, we used diameter of the projection as it leaves the soma as our main criterion for distinguishing axons and dendrites. Location of synapses was assigned to somata, proximal dendrites, or distal dendrites of GnRH neurons.
Tuning the passive model with the genetic algorithm
Data were gathered from 11 different cells. These data were analyzed for input resistance and an average short-cip response was generated for each cell. These average responses were computed from both the positive and negative cip and were scaled by the magnitude of the injected current. The resulting averaged, normalized cip response was fit with a three-exponential function, to extract three time constants (0, 1, and 2).
The response of a neuron to an injection of charge, from either an experimental current injection or a synapse, will consist of two main components. First, there will be a passive response, as the cell membrane changes polarization due to the change in charge, and current leaks out of the cell through voltage-independent leak channels, and charge is redistributed throughout the cell by axial conduction. The second, active part of the response is due to the activation and inactivation of voltage-dependent conductances. Thus, in building a model of a cell, it is important to first simulate the passive response of the cell as accurately as possible. Voltage responses of passive neurons to current injection pulses were simulated in GENESIS and were compared with the voltage responses of the corresponding living neuron. The simulated voltage response in each model GnRH neuron was matched to its electrophysiological data using an algorithm that simultaneously finds the best fit for the parameters membrane capacitance, Cm, membrane resistance, Rm, and axial resistance, Ra.
Due to the large parameter space of multicompartmental models, automated parameter optimization algorithms can be extremely useful for finding parameter sets producing desirable model activity. Vanier and Bower (13) demonstrated that genetic algorithms are among the most effective parameter-search methods for large parameter spaces (up to 23 parameters). In brief, the genetic algorithm works by creating an initial population of parameter sets with randomly chosen values (within defined bounds) for each parameter in a set. Each parameter set is rated by comparing the model activity with the parameter set against the defined activity in the living neuron. In an evolution-inspired fashion, new parameter sets are created by breeding the highest-rated parameter sets. The new offspring are then rated, and the best rated sets are bred to produce new offspring. The breeding continues throughout a set number of generations to converge upon a best parameter set. The quality of the fit between the modeled neuron and the same neuron during recordings was determined by comparing the negative root mean square of the simulated and real voltage responses. A "good" fit requires less than 1% error between the simulated voltage responses and the voltage response in the living neuron.
Creating the active model
The genetic algorithm was used to optimize passive parameters for all 11 measured and reconstructed cells. The two model cells with the best passive fits were used to build active models. These GnRH neurons were also representative of the two major morphological types we identified (see Results). Once the passive properties of each model were determined through optimization with the genetic algorithm, active voltage-gated channels, modeled with Hodgkin-Huxley equations, were added to soma and axon compartments. The kinetics of the channels were adapted from published models of GT1–7 cells (14, 15). The models included fast sodium (Na), delayed rectifier potassium (Kdr), and L-type calcium (CaL) conductances (14, 15, 16, 17).
For this conductance-based active model, cell membrane potential is calculated as:
(1)
is the passive membrane leakage current, and Iinj is injected current. Each of the ionic currents is described by an equation of the form: Ii = gi(Vm,t)(Vm – Ei), where the ionic conductance, gi, is a specified function of membrane potential and time, and the reversal potential, Ei, depends on the ionic species. The ionic conductances, in the formalism of Hodgkin and Huxley, were described in terms of a maximum conductance for each channel type (Na, (Kdr, and (CaL), and one or more gating variables. The gating variables model the activity of activation gates (m for Na and CaL and n for Kdr) and inactivation gates (h for Na). For this model, the three conductances were described by the equations:
(2)
(3)
(4)
The exponents on the gating variables, m, h, and n indicate that the Na conductance has three activation gates and one inactivation gate, and the K delayed rectifier and L-type Ca conductances have no inactivation gates, and 4 and 2 activation gates, respectively. The maximum conductance, i, represents the single channel conductance times the channel density. It is these conductance values that are tuned to make the model duplicate measured electrophysiological activity.
The kinetics of a particular type of channel are modeled by specifying the functional dependence of the gating variables on membrane voltage and time. The voltage dependence for each gating variable, x {m, h, n} is described by a steady-state value, x(V), and a time constant, x(V). The time course of the change in activation or inactivation for a change in potential from V1 to V2 is:
(5)
The voltage dependence of the steady-state gating variables and time constants are all modeled in the forms:
(6)
And
(7)
(8)
The values for the kinetic parameters s (= +1 or –1), ssx, dtx, (seconds), Vx, Kx, Vtx, and Ktx (mV) were all taken from the published values of (14, 15), with the exception of the fast sodium, where the half-activation potentials, Vm and Vh, were decreased by 14.6 mV, to more accurately duplicate spike threshold and amplitude. All of the time constants were divided by a rate factor of 5 to compensate for the fact that the data on which the models in (14, 15) were based were taken came from cells at 25 C, whereas the present study was based on brain slices measured at 32 C (see Ref.18 for an explanation of this temperature scaling of rate constants). Table 1 summarizes the values of the kinetic parameters used in this model.
The relative densities of these three types of channels (Na, (Kdr, and (CaL), were adjusted separately in the soma and axon, to duplicate measured active responses to current injection. The channel densities were tuned to simultaneously match measured resting membrane potential, and response to the three types of current injection described above. Each model neuron was tuned to duplicate the physiological responses of the corresponding living cell before being used to test simulated synaptic inputs.
Making synapses for simulated inputs to virtual GnRH neurons
Multiple synaptic phenotypes have been identified on GnRH neurons. We focused on the impact of excitatory inputs because whole cell recordings indicate GnRH neurons have low intrinsic rates of firing (9, 10), and therefore most observed activity will likely be the result of excitatory input. Glutamate is a major excitatory neurotransmitter in the hypothalamus (19) and has been implicated in control of GnRH secretion (20). Although GABA has also been reported to be excitatory in GnRH neurons (21), this remains controversial (22). Thus, we focused on glutamatergic inputs. Characteristics of glutamate currents in GnRH neurons have been well defined (9, 23, 24). The time course of AMPA inputs, which are similar in GnRH neurons to other neurons, was defined using a dual exponential function
(9)
with a 1 and 2 of 0.5 and 1.2 msec, respectively.
Bifurcation analysis
To aid in interpretation of model results, a bifurcation analysis of the model was performed. For this purpose, a single-compartmental version of the model was developed; this reduced model was then simulated using the software XPPAUT, developed by G. B. Ermentrout and available at ftp://ftp.math.pitt.edu/pub/bardware. A copy of the XPPAUT file containing the model is available from the authors upon request. The conductance densities in the soma of the reduced model were tuned to match active electrophysiological responses as described for the full model above.
Bifurcation analysis produces a diagram relating the voltage response of the model cell to applied current, thus providing a visualization of the threshold amount of injected current required to generate an action potential in the model. The response of a cell to injected current may vary depending on the state of the cell before current injection; all possible responses (such as remaining at rest or spiking) should appear in the bifurcation diagram.
Results
Morphological properties of GnRH neurons
Table 2 indicates the morphological characteristics of biocytin-filled GnRH neurons. As previously demonstrated, GnRH neurons typically have one main dendrite. However, 36% of dendrites branched, giving rise to secondary dendrites which could be relatively long [average: 786.4 (±169.8) μm]. Dendrites typically branched in more distal locations [414.4 (± 164.9) μm], but the location of branch points was highly variable. The main dendrite was on average 510.0 (±89.6) μm in length.
The anatomical locations (panels A and D), biocytin reconstructions (panels B and E) and GENESIS renderings (panels C and F) of two modeled GnRH neurons are shown in Fig. 1. Approximately half of GnRH neurons had dendrites that arched and sent projections back in the direction of somatic fields (see Fig. 1B). Most of these dendrites closely followed the trajectory of the GnRH axon. The branching cell in Fig. 1, B and C, was one of the two used in active modeling. Other GnRH neurons exhibited the morphology typically reported in this population of cell (Fig. 1, E and F). This second, bipolar cell was the other made into an active model for this study.
Figure 2, A and E, compares the fit of passive voltage response in living and modeled bipolar (A) and branching (E) GnRH neurons to short current injection pulses (Fig. 2, B and F). Figure 2, C and G, shows representative response of a living GnRH neuron (black trace) to a current injection pulse of +0.5 nA for 0.5 msec (Fig. 2, D and H) before the application of the pharmacological solution (to render the neuron passive) and the corresponding voltage response of the model (gray trace). In response to this current injection, living GnRH neurons exhibited one to two action potentials similar to published results in GnRH neurons (24, 25). Simulated neurons were tuned to exhibit similar responses. Both models were tuned to match the same measured response (from a third cell). Because the total area of the bipolar cell was less than that of the branching cell, the bipolar cell had less total capacitance. The total capacitance of the model bipolar cell was 29.75 pF, compared with 58.79 pF for the branching cell. This meant that a smaller current input was required to generate the same voltage response in the smaller cell (0.261 nA compared with 0.45 nA). Living and simulated neurons exhibited relatively low spontaneous rates of firing, consistent with findings in hypothalamic GnRH neurons in slices using whole cell recordings (9, 10).
The bifurcation diagram for the single-compartment model is shown in Fig. 3A. In the graph, injected current increases along the horizontal axis and membrane potential varies along the vertical axis. For a given amount of injected current, the diagrams show all mathematically possible responses; a response that may actually be observed is referred to as stable. Solid lines indicate stable resting membrane potentials. Filled circles mark the maximum and minimum voltages of spikes that may be observed. Dashed lines and open circles represent unstable membrane potentials or spikes, respectively. These unstable responses are of interest for the mathematical analysis but would not be observed experimentally or in numerical simulations of the model.
These diagrams show that the model cell would not fire in the absence of injected current. With no injected current, the model gives a resting membrane potential of –49.8 mV. With current injected at the soma, Fig. 3A shows that the resting membrane potential rises from –49.8 mV to –36.5 mV as the current is steadily increased from 0 to 0.019 nA. As the injected current increases beyond 0.0192 nA, the cell begins spiking. Injected currents greater than 0.408 nA result in sustained depolarization.
To understand the impact of relatively long axons and dendrites on firing, we compared responses to current injection in single and multicompartment models. It is difficult if not impossible to generate full bifurcation diagrams from compartmental models with hundreds of compartments. Therefore, we studied the single compartment model generated in XPP (from which the bifurcation diagram was derived), a single compartment model with identical parameters constructed in Genesis, and the multicompartment Genesis model of the branching cell (e.g. a model that includes the dendrite and axon). The single compartment XPP and Genesis models are nearly identical in their firing rates in response to current injection. In contrast, in the full morphology model the general shape of the response is similar but the spike rates are lower. Moreover, the range of injected currents that lead to spiking is narrower in the multicompartment model.
Single synaptic inputs and spike initiation
To assess the effect of attenuation of synaptic input with increasing distance from the soma, a single simulated AMPA synapse, with the mathematical form described above, was placed first in the soma compartment of each model, and then at progressively more distal compartments until the postsynaptic response was insufficient to initiate an action potential. For a single synapse at the soma, the minimum unitary conductance sufficient to initiate an action potential was 2360 pS for the bipolar cell, and 3170 pS for the branching cell. Once again, the smaller bipolar cell’s lower total capacitance made it more responsive to smaller synaptic conductance amplitudes (postsynaptic currents). This threshold conductance amplitude was insufficient to generate spikes from any distal compartments past than the soma (bipolar cell) or 22 μm from the soma (Fig. 4, A and D). With twice the threshold conductance amplitude, a single synaptic event was able to initiate action potentials at distances up to 257 μm from the soma on the bipolar cell, and for the branching cell, at up to 329 μm from the soma on the short dendrite branch, and 245 μm out on the longer branch (Fig. 4, B and E). With three times the threshold value (7080 pS for the bipolar cell and 9510 pS for the branching cell), a single input was effective up to 299 μm (bipolar cell) and 358 μm on the short branch, and 275 μm on the long branch of the branching cell (Fig. 4, C and F). These distances compare with total measured dendrite lengths of 381 μm for the bipolar cell and 1236 μm for the longest branch of the dendrite of the branching cell. Thus, even relatively large synaptic inputs are insufficient to drive spiking when applied at a majority of the distance along the total dendrite length.
Ongoing synaptic input
In living neurons, synaptic inputs are not single strong pulses but consist of multiple inputs from one or more presynaptic cells. To simulate this, five independent inputs, each firing at an average rate of 20 Hz, were placed at varying distances from the soma. The actual number of inputs is important only in the context of total input frequency because total input frequency is the number of synapses multiplied by input frequency of each synapse. Activation times for inputs were based on a Gaussian distribution of activation intervals. The maximum conductance of each of the randomly firing synapses was set at 1250 pS. This replicates a 70 pA postsynaptic current, some of the largest currents that are found in GnRH somata during bath application of glutamate (23). Figure 5 shows a representative response of the branching model GnRH neuron to identical synaptic trains applied at various distances from the soma. When synapses were placed on the soma, firing was induced at an average rate of 10.1 Hz (5A). This rate of firing is consistent with our findings using dynamic current clamping to apply simulated AMPA inputs to living GnRH neurons. Application of this conductance profile (5 inputs at 20 Hz each for a total input of 100 Hz with unitary conductance of 1250 pS/input) resulted in a firing rate of about 9 Hz (24). Thus, there is good agreement between the response to synaptic trains in our virtual GnRH neurons in the present study and living GnRH neurons in an earlier study (24). The rate of firing decreased with increasing distance from the inputs to the soma. The induced spike rate fell less than 2 Hz for inputs 328 μm (5B) from the soma on the short branch, and 245 μm (5C) from the soma on the long branch of the dendrite, respectively. The bipolar model cell showed similar response, although with higher induced spiking frequencies for more proximal compartments, due to the lower total capacitance.
Figure 6 summarizes the impact of dendrite length on firing rate as assessed by simulated somatic voltage. As discussed previously, the smaller bipolar cell exhibits a larger induced spike frequency for proximal inputs, but for compartments more distal than 125 μm, both models showed similar response. Thus, even a strong synaptic stimulus loses its ability to drive high frequency firing at a distance that is less than the average dendritic length.
Discussion
The present study is the first to construct multicompartmental models of hypothalamic GnRH neurons. Two earlier studies developed so-called point models of GT1–7 cells (14, 15). A point model differs from a multicompartmental model because it treats the neuron as occupying a single point location in space. Thus, the behavior is completely independent of neuronal morphology. In contrast, a multicompartment model incorporates morphological aspects of the neurons of interest.
To understand their fundamental behavior, we reduced the models to a single compartment and used geometric dynamical systems methods to systematically analyze the activity seen in the reduced model. We also used the multicompartmental models with full, defined morphology to understand how synaptic input to the dendrite of GnRH neurons would control firing.
Surprising new insights into the morphology of GnRH neurons
The morphology of GnRH neurons is generally described as simple bipolar with a vertical orientation in the hypothalamus. The recent efforts of Campbell et al. (5) using coronal slices have challenged this view regarding the morphology of GnRH neurons. In the present study, we used hemisagittal slices. Thus, we can provide additional, important insight to this emerging picture. First, as noted by Campbell et al. (5), half of primary dendrites exit a coronal slice, thereby preventing measurement of their total length. In our hemisagittal slice orientation, the average length was longer than the average length reported in coronal slices (500 vs. 300 μm). Moreover, three of our filled dendrites had lengths that exceeded 1200 μm in length, somewhat longer than the maximum lengths reported by Campbell et al. (5). However, even in our slice orientation dendrites frequently extended beyond the plane of the slice. Thus, both studies are in agreement that dendrites of GnRH neurons are much longer than previously appreciated and likely extend into distal hypothalamic and perhaps extrahypothalamic regions.
Our findings support the observations that GnRH dendrites change directions after leaving the cell body (5), and we can extend this observation by demonstrating the trajectory of these dendrites. Using the hemisagittal slice orientation, our findings provide evidence that dendrites of about half of GnRH neurons closely follow the path of the axon. These dendrites may project to the median eminence. Several neurotransmitters have been suggested to regulate GnRH secretion at the level of the median eminence by acting on axons or nerve terminals. Synapses, however, are not generally identified on these structures. Our findings suggest that regulation of GnRH secretion at the level of the median eminence could occur through synapses on the dendrites of GnRH neurons.
Understanding the firing behavior in the reduced model
Bifurcation analysis of the reduced model predicts that sustained firing occurs only in the presence of continuously injected current of sufficient strength. This finding suggests that the mechanisms leading to repetitive firing in GnRH neurons may differ from those in other neurosecretory neurons.
Based on bifurcation analysis, sustained firing in GnRH neurons appears to depend on currents arising from extrinsic sources (i.e. synaptic inputs). Extrinsic current destabilizes the resting membrane potential and the GnRH neuron fires action potentials. These findings are consistent with dynamic-clamping experiments that indicate on-going synaptic input is required to maintain firing in GnRH neurons (24). In contrast, cultured GnRH neurons and GT1–7 cells exhibit spontaneous firing and GnRH secretion (1, 25). The spontaneous release of GnRH pulses in culture has formed the basis for the theory that intrinsic cellular properties can account for intermittent firing and pulsatile GnRH release. However, intrinsic properties can be significantly altered in culture. In GT1–7 cells, intervals between GnRH pulses and Ca2+ oscillations decrease from 80–21 min after 30 d in culture (25). Culturing has been shown to induce up-regulation of ion channels and alterations in firing properties such that firing patterns shift from irregular to burst firing despite the absence of bursting behavior in the intact ganglion (26, 27). Moreover, two recent studies indicate expression of calcium channels in hypothalamic GnRH neurons in short-term culture is dramatically lower and differs in phenotype from calcium channels expressed in GT1–7 cells and primary GnRH neurons in long-term culture (16, 17). Therefore, based on the present findings, it appears that the cellular mechanisms accounting for pulses from GT1–7 cells and cultured neurons may not be the same as those in hypothalamic GnRH neurons.
The above consideration notwithstanding, the model of LeBeau et al. (15) faithfully reproduces the firing patterns of GT1–7 cells. The GT1–7 cell model of LeBeau et al. (15) had several currents in addition to the ones used in the present models. Other currents in the GT1–7 model, in particular the store-operated Ca2+ channel (ISOC) and the nonselective inward cation current (Id), would be expected to have an effect on intrinsic spiking, bursting, and their inclusion in simulations of GT1–7 cells resulted in models that replicated experimental findings. Moreover, these conductances may be important in several modulatory responses. For our study of the mechanisms of spike generation and the effects of synaptic input, only the three strongest currents in the GT1–7 model were used. Additional currents were not necessary to duplicate the measured electrophysiology in the cells studied here, and were therefore not included. It should be emphasized, however, that the activity of the GT1–7 cells as described by LeBeau et al. (15) is different from that of the cells measured in this study, and thus the intrinsic currents may be expected to differ as well. In this regard, GT1–7 cells are derived from a clonal cell line, and their behavior (and conductances) may represent those of a specific subpopulation of GnRH neurons in the hypothalamus. Such differences in expression of ionic conductances would account for functional heterogeneity in hypothalamic GnRH neurons (28) and underscores the need for extensive study of conductance phenotypes expressed in hypothalamic GnRH neurons.
Understanding the impact of synaptic inputs in the multicompartmental model
The long dendrites of GnRH neurons potentially provide an extensive substrate for synaptic integration. Classical cable theory, however, predicts attenuation of signals in dendrites due to membrane capacitance and leak resistance (29, 30, 31, 32). In theory, attenuation of fast synaptic currents would become near-total in extended thin dendrites, like those of GnRH neurons. Experimental approaches such as local electrical stimulation to activate putative inputs to dendrite would provide only ambiguous results. Experimentally, one can stimulate fibers in the region of a dendrite, but the actual location of the presynaptic nerve terminal on the GnRH neuron cannot be determined. Thus, in the present study, we used experimentally based models of GnRH neurons to directly test the impact of synaptic input to specific regions of the dendrite. We determined where, on the extended dendrite, synapses could reside and still drive the repetitive firing that appears to be required for neuropeptide secretion (33). The pattern and frequency of firing that releases GnRH is unknown, but vasopressin neurons fire phasically with mean rates during bursts of 5–15 Hz during periods of hormone secretion (33). Moreover, at least 2 Hz is required to release enough GnRH to evoke postsynaptic events (34). During release of LH, single units extracted from multiunit activity volleys in the primate hypothalamus exhibit firing rates of 2–20 Hz (35). Thus, at least 2 Hz firing is likely required to drive GnRH release. In hypothalamic slices (9, 10), dispersed somata of GnRH neurons (23) and in cultures of GnRH neurons from the1) firing frequencies of at least 2 Hz have been defined. Therefore, we looked for dendritic locations where activation of synapses would result in firing frequencies of at least 2 Hz.
We focused on determining the impact of AMPA-type glutamatergic inputs because AMPA-type receptors appear to be the dominant form of glutamatergic receptors in GnRH neurons (9). Moreover, AMPA-type ligands are more effective than NMDA ligands at inducing repetitive firing in pharmacological experiments on GnRH somata (23). Other synaptic phenotypes have been identified on GnRH neurons (4, 36), some of which may provide additional excitation. For example, GABAergic inputs have been suggested to be excitatory in GnRH neurons (21). The reversal potential for GABA excitatory inputs in (21) was –36.5 mV. In contrast, the reversal potential for AMPA inputs is 0 mV. Thus, the electrochemical driving force (Vm–Erev) is larger for AMPA-mediated currents than the driving force on GABAergic currents by 36.5 mV. The impact of these other synaptic phenotypes remains to be determined. However, examining the impact of AMPA-type inputs provides the best case scenario for independent control of firing by excitation for the two major fast neurotransmitter phenotypes in the hypothalamus (e.g. glutamate and GABA).
We found in simulations of morphologically reconstructed GnRH neurons that AMPA-type excitatory post-synaptic currents (EPSC) and resulting excitatory post-synaptic potentials (EPSP) were significantly attenuated over a substantial portion of the dendrite length. Accordingly, even relatively high frequency, large amplitude currents derived from synaptic input could not drive repetitive firing when placed at dendritic locations more distal than less than half the actual observed dendrite length. The finding that synaptic input to a majority of the dendrite has little to no effect on somatic voltage raises significant issues for synaptic processing in GnRH neurons. The number of synapses on GnRH somata is limited. In male rats, there are two to 12 synapses on the somata of GnRH neurons when examined at the electron microscopic level using serial thin sections (2). Earliest reports using randomly selected sections for quantification suggested that most GnRH neurons had either no synapses (37) or less than one synapse/cell (3). The more recent analysis using serial sections, however, revealed that all GnRH neurons have at least two synapses, with an average of about seven synapses on somata (2). Thus, at the level of the somata, GnRH neurons are sparsely innervated suggesting that dendrites may be the principle sites of synaptic regulation. In mice, synaptic inputs to GnRH neurons appear to be qualitatively similar to other models (4, 38) with approximately 30% higher synaptic density on the proximal dendrite relative to the cell body. Moreover, synaptic density on dendrites increases during the breeding season in sheep when GnRH secretion is elevated (6). However, the present study indicates that only synapses located on somata and very proximal dendrites are capable of controlling somatic voltage. Thus, the findings of the present study indicate that, although the dendrite may receive extensive synaptic input, the morphology of the dendrite limits synaptic efficacy in the context of control of firing.
The finding that distal dendritic synapses do not alter firing calls into question the physiological significance of changes in input to the dendrite. It should be noted, however, that our model makes the assumption that the dendrite of the GnRH neuron is passive. This assumption is consistent with the dominant perspective in neuronal signaling, namely that action potentials are initiated in the soma/axon hillock region where voltage-sensitive sodium channels are clustered. This classical view of neuronal processing considers dendrites largely as passive conduits that relay changes in voltage from synapses to the axon hillock. In some populations of neurons, however, dendrites express active conductances, and in particular voltage-gated sodium channels. This endows dendrites with the ability to exhibit action potentials generally through back propagation of action potentials initiated at somata (39, 40, 41 ; see Ref.42 for review). Recent studies have indicated that in some circumstances, action potentials can be initiated in the dendrite through activation of voltage-sensitive sodium channels by excitatory inputs (43, 44, 45, 46, 47). If this were the case, the dendrite could be a potential site of synaptic integration. Additionally, dendrites with active properties could underlie synchronized firing in GnRH neurons via a common excitatory synaptic input and, as suggested by others, direct electrical communication between dendrites (48). It is currently unknown whether the dendrites of GnRH neurons possess active properties. This consideration notwithstanding, the findings of the present study indicate that purely passive integration is unlikely to be a suitable mechanism for the processing of synaptic input over much of the dendrite length.
Acknowledgments
We thank Dieter Jaeger for permitting the use of his laboratory for the electrophysiological components of this work. We thank Dr. Arthur Sherman for providing his ordinary differential equation files from 14 and 15 to K.J.S. during her participation in Methods in Computational Neuroscience at the Marine Biological Laboratory at Woods Hole.
Footnotes
This work was supported by HD-045436 (to K.J.S.) Preliminary data collection in coronal slices was supported by NS-010355. Support for the course at Woods Hole was provided by MH-062204.
A preliminary report of these data were presented in abstract form at the Annual Meeting of The Endocrine Society Meeting in 2004.
First Published Online December 22, 2005
Abbreviations: ACSF, Artificial cerebrospinal fluid; AMPA, (S)--amino-3-hydroxy-5-methyl-isoxazolepropionic acid; CaL, L-type calcium; GABA, -aminobutyric acid; GFP, green fluorescent protein; Kdr, delayed rectifier potassium.
Accepted for publication December 9, 2005.
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