Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents

The profile of transmembrane-channel expression in neurons is class dependent and a crucial determinant of neuronal dynamics. Here, a generalization of the experimentally verified exponential integrate-and-fire model is introduced that includes biophysical, nonlinear gated conductance-based currents, and a spike shape. A Fokker-Planck-based method is developed that allows for the rapid numerical calculation of steady-state and linear-response properties for recurrent networks of neurons with gating-variable dynamics slower than that of the voltage. This limit includes many cases of biological interest, particularly under in vivo conditions of high synaptic conductance. The utility of the method is illustrated by applying it to two biophysically detailed models adapted from the literature: an entorhinal layer-II cortical neuron and a neuron featuring both calcium-activated and voltage-activated spike-frequency-adaptation currents. The framework generalizes to networks comprised of different neuronal classes and so will allow for the modeling of emergent states in neural tissue at significantly increased levels of biological detail.

Figure 1

Steady-state properties of a generalized EIF Model (GEM) with a nonlinear depolarization-activated hyperpolarizing current. (A)–(C) Example with a tonic synaptic drive $g_{s0}=2g_L$, $E_{s0}=-30\text{mV}$ ($g_{e0}=1.14g_L$, $g_{i0}=0.86g_L$), and fluctuations $\sigma =4\text{mV}$. (A) Voltage $V\left(t\right)$ and gating variable $x\left(t\right)$ [simulations of Eqs. (7, 8)] with average firing rate of $\sim 18\text{Hz}$ for the simulations. The probability density (circles, simulation) is shown in the inset together with $x_{\infty }\left(V\right)$. (B) Voltage profiles of ${\tau }_x\left(V\right)$ and $x_{\infty }\left(V\right)$ shown with examples of the weighting function $P_0\left(V;x_0^{in}\right)/{\tau }_x\left(V\right)$ for $x_0^{in}=0.14,0.35,0.9$. (C) Two schemes for finding $x_0$. Solid line shows $x_0^{out}$ [Eq. (17)] for the full range $x_0^{in}=0\to 1$ with the intersection $x_0^{out}=x_0^{in}=0.35$ giving the self-consistent solution. This is identical to the simulational average (circle) to two decimal places. An alternative iterative scheme [dashed line, Eq. (18)] is also shown, with the first two points (squares, $x_0^{in}=0.9,0.14$) and fixed point (star, five iterations $x_0^{in}=0.35$). These values were used for the example weighting functions of panel (B). The theoretical probability density $P_0\left(V;x_0=0.35\right)$ is plotted in the inset of panel (A) and agrees well with the simulation (circles) with the theoretical steady-state rate $r_0=18.1\text{Hz}$ also close to the simulational value. (D) Steady-state gating variable $x_0$ [solutions of Eq. (17)] and (E) corresponding firing rates [Eqs. (15, 16)] for a range of $E_{s0}$ with fixed $g_{s0}=2g_L$. Theory (lines) and simulation (symbols) are compared for three fluctuation strengths $\sigma$. Filled symbols correspond to the example in panels (A)–(C). Other model parameters are provided in Appendix .

Figure 2

Response of a population of GEM neurons to modulated synaptic conductance. (A) Simulations of voltage, population-averaged rate, gating-variable, and period-averaged gating-variable for a sinusoidally modulated excitatory conductance $g_{e1}=0.057g_L$ at a frequency of 10 Hz [all other parameters are shared with the steady-state example in panels (A)–(C) of Fig. 1]. (B) Amplitude and phase of the modulated gating ${\widehat{x}}_1$ [solid line, theory (42); symbols, simulation; filled symbol 10 Hz example of panel (A)]. (C) Corresponding amplitude and phase of the firing-rate modulation [Eq. (38)]. A firing-rate resonance at $\sim 13\text{Hz}$ and phase zero at $\sim 5\text{Hz}$ are visible, but absent in the rate modulation of an equivalent neuron with no gating-variable dynamics [dashed lines, Eq. (38) with ${\gamma }_x=0$]. (D) The firing-rate filter $A_e\left(t\right)$ for excitatory synaptic conductance modulation derived from inverse Fourier transforming $r_1\left(\omega \right)$ [see Eq. (44)]. The negative tail is characteristic of the slow negative feedback generated by this particular voltage-activated current. (E) Population response to patterned excitatory synaptic drive [solid line, theory Eq. (45); gray histogram, simulations]. Note the sag or overshoot response to step changes in conductance. Parameters not provided in the caption to Fig. 1 can be found in Appendix .

Figure 3

A recurrent network of inhibitory GEM neurons with voltage-activated currents. (A) Construction for finding the steady-state network rate $r_0$ (for an example $g_{e0}^a=1.14g_L$ and $g_{i0}^a=0.75g_L$). Network rate and combined inhibitory conductance $g_{i0}$ are found at the intersection of the population rate curve parametrized by $g_{i0}$ [Eqs. (15, 16)] and the mean-field rate curve [Eq. (48)]. The intersection agrees well with the rate (circle symbol) from a simulation of the recurrent network. (B) Simulated voltage and gating variables for a neuron in the network as well as the combined inhibition. Dashed lines are the theoretical predictions for $x_0$ and $g_{i0}$. (C) The amplitude and (D) the phase of the firing-rate modulation ${\widehat{r}}_1$ [bold line, Eq. (56); symbols, simulations] for a modulated excitatory conductance ${\widehat{g}}_{e1}^a=0.057g_L$. Rate modulation for a population [dashed line, Eq. (56) with ${\widehat{\gamma }}_{i1}=0$; see Fig. 2C] and an inhibitory network of neurons without voltage-activated currents [dotted line, Eq. (56) with ${\gamma }_x=0$] are also plotted for comparison. The network coupling $c_i=0.59g_L$ and fluctuations $\sigma =4\text{mV}$ were chosen to give a steady state equivalent to that in Fig. 1.

Figure 4

Two biophysical models. (A)–(F) Entorhinal cortex layer-II neuron with persistent sodium and $I_h$ currents. (A) Steady-state rate as a function of synaptic potential $E_{s0}$ (with $g_{s0}=2g_L$) for two fluctuation strengths (lines, theory; symbols, simulations). (B) Voltage dependence of time constants and activation plotted with the steady-state density $P_0$ (line, theory; gray histogram, simulation) for the case $E_{s0}=-60\text{mV}$, $g_{e0}=0.29g_L$, $g_{i0}=1.71g_L$, and $\sigma =8\text{mV}$ [filled symbol in panel (A)]. (C) Voltage and gating-variable time courses for the same case (dashed lines, steady-state approximations $s_0,f_0$). (D) Firing-rate response to modulated excitatory synaptic conductance $\left(g_{e1}=0.057g_L\right)$ for the example, showing a resonance near 10 Hz. (E) Real-time filter corresponding to the inverse Fourier transform of panel (D). A weak undershoot is perceptible. (F) Response to patterned excitatory stimulus [thin line, with $g_{e1}\left(t\right)/g_L$ three times that of Fig. 2E] showing good agreement between theory (bold line) and simulation (gray histogram). (G)–(L) Modified Traub-Miles neuron with voltage-activated $I_M$ and calcium-activated $I_{mAHP}$ potassium currents. (G) Steady-state voltage as a function of $E_{s0}$ (with $g_{s0}=2g_L$) for two fluctuation strengths. (H) Time constants and activation profiles with the steady-state density $P_0$ for the case $E_{s0}=-30\text{mV}$, $g_{e0}=1.14g_L$, $g_{i0}=0.86g_L$, and $\sigma =10\text{mV}$ [filled symbol in panel (G)]. (I) Voltage and gating-variable time courses for the same case (dashed lines, steady-state approximations $w_0,{\left[{\text{Ca}}^{2+}\right]}_0/\left(30+{\left[{\text{Ca}}^{2+}\right]}_0\right)$). (J) Firing-rate response to modulated excitatory synaptic conductance $\left(g_{e1}=0.057g_L\right)$ for this case, showing a pronounced resonance near 40 Hz. (K) Corresponding real-time filter exhibiting a visible undershoot. (L) Response to a patterned excitatory stimulus [$g_{e1}\left(t\right)/g_L$ 1.5 times that of Fig. 2E]. All other model parameters can be found in Appendix .

Figure 5

A population of GEM neurons with spike shape [Eq. (B3)] of width ${\tau }_{ref}=5\text{ms}$. (A) Steady-state rate as a function of excitatory-inhibitory balance $E_{s0}$ with $\sigma =4\text{mV}$ and $g_{s0}=2g_L$ [bold line, theory; symbols, simulation; dashed line, nonrefractory case of Fig. 1E]. (B) Steady-state density when $E_{s0}=-30\text{mV}$ (bold line, theory; symbols, simulation). A voltage time course is shown below in which the spike shape is discernible. (C) The rate amplitude and (D) the phase in response to modulated excitatory conductance [bold line, theory; symbols, simulation; dashed line, nonrefractory case of Fig. 2C]. Apart from the spike shape, all other parameters are identical to the corresponding cases in Figs. 1, 2.