Fokker-Planck description of conductance-based integrate-and-fire neuronal networks

Steady dynamics of coupled conductance-based integrate-and-fire neuronal networks in the limit of small fluctuations is studied via the equilibrium states of a Fokker-Planck equation. An asymptotic approximation for the membrane-potential probability density function is derived and the corresponding gain curves are found. Validity conditions are discussed for the Fokker-Planck description and verified via direct numerical simulations.

Figure 1

(Color online) Black: gain curves $m$ as a function of $f\nu$ in the mean-driven limit. Gray (red online) dashed line: the locations of the saddle-node bifurcation points on the gain curves. The parameter values are $B=14/11=1.2727$, and $\tau =1$. The reversal and threshold potentials satisfy Eq. (2). The coupling strengths along the gain curves are (left to right) $S=0.3$, $S=\tau \text{ln}B=0.2412$, $S=0.2$, $S=0.1$, and $S=0$.

Figure 2

The $f\nu$ coordinate, $\overline{f\nu }$, of the firing-rate saddle-node bifurcation as a function of $S/\tau$. The parameter values are $B=14/11=1.2727$ and $\tau =1$. The reversal and threshold potentials satisfy Eqs. (2).

Figure 3

(Color online) Gain curve: the firing rate $m$ as a function of the driving strength $f\nu$. The reversal and threshold potentials satisfy Eq. (2). The parameter values are $\tau =1$, $S=0.1$, and $\lambda =1$, where $pN=\lambda /f$ and ${\delta }_s=0.0002$. The strength of the external input spikes $f$ varies from curve to curve. From left, the values of $f$ are 0.1, 0.05, 0.025, 0.005, 0.001, 0.0001, and 0, respectively. Note that, along the left three gain curves (dashed; red online), $f$ does not satisfy the small-jumps condition (14) and the Fokker-Planck equation (20) becomes therefore progressively less valid with increasing $f$ and with proportionally decreasing $pN$. The corresponding minimal numbers of jumps needed for a neuron to reach $V_T$ beginning at ${\epsilon }_r$, computed from Eq. (14), are $\sim 2$, 4, 8, 40, 200, and 2000, respectively. The rightmost gain curve (gray; green online) was plotted using the parametrization (28) in the mean-driven limit ($f=0$, $pN\to \infty$, $\nu \to \infty$, $f\nu$ finite). Inset: convergence of the gain curves to the mean-driven limit gain curve near the upper turning point as $f$ decreases and $pN$ increases proportionally.

Figure 4

(Color online) Probability density functions $\rho \left(v\right)$ at specific points on the $m-f\nu$ gain curve. The reversal and threshold potentials satisfy Eq. (2). The parameter values are $\tau =1$, $f=0.001$, $S=0.1$, $pN=1000$, and ${\delta }_s=0.0002$. (a) Specific points on the gain curve. (b) $\rho \left(v\right)$ at these points. The functions $\rho \left(v\right)$ were computed via the asymptotic formula (40). [Note that $\rho \left(v\right)$ computed numerically via Eqs. (30, 31, 35) instead of Eq. (40) appear identical.] Inset: the boundary layer at $v=V_T$ of the probability density functions $\rho \left(v\right)$ corresponding to the location “D” on the high-firing part of the gain curve as shown in black dashed line. (Note the scale on the $v$ axis.) In gray (green online) is shown $\rho \left(v\right)$ in the mean-driven limit, described by Eq. (24).

Figure 5

(Color online) Comparison between a gain curve (black dashed) computed from the steady-state Fokker-Planck equation as described in Sec. 7 (displayed on left of Fig. 4) and its approximations in the fluctuation-driven (light gray; green online) and mean-driven (dark gray dash-dotted; red online) regimes, computed via Eqs. (48, 28), respectively. The parameter values along the computed gain curve are $\tau =1$, $f=0.001$, $S=0.1$, $pN=1000$, and ${\delta }_s=0.0002$. The reversal and threshold potentials satisfy Eq. (2).

Figure 6

(Color online) Comparison between the probability density function $\rho \left(v\right)$ computed using the uniform asymptotic approximation (40) of the steady Fokker-Planck solution and the Gaussian approximation (49), which is valid in the fluctuation-driven regime. The parameter values are $\tau =1$, $f=0.001$, $S=0.1$, and $pN=1000$. The reversal and threshold potentials satisfy Eq. (2). Inset: the value of the external driving strength $f\nu$ is the same as at the location “B” on the left panel of Fig. 4.

Figure 7

(Color online) Comparison between the probability density function $\rho \left(v\right)$ computed using the uniform asymptotic approximation (40) of the steady Fokker-Planck solution and the approximation (51), which is valid in the mean-driven regime. The reversal and threshold potentials satisfy Eq. (2). The parameter values are $\tau =1$, $f=0.001$, $S=0.1$, and $pN=1000$. Note that the mean-driven approximation curve is only normalized to $O\left(1\right)$ and is missing an $O\left(q^2/a\right)$ amount, as explained in the text. Inset: the value of the external driving $f\nu$ is the same as at the location “C” on the left panel of Fig. 4.

Figure 8

(Color online) Comparison of gain curves evaluated theoretically (black, dashed) as in Sec. 7 and shown in Fig. 3 and gain curves computed via numerical simulations (gray; green online). The reversal and threshold potentials satisfy Eq. (2). The parameter values are $\tau =1$, $S=0.1$, $\lambda =1$, where $pN=\lambda /f$. The strength of the external input $f$ assumes the values 0.1, 0.025, 0.005, and 0.002, respectively, on the gain curves from left to right. In the simulations, $N={10}^4$, except on the bottom half of the gain curve corresponding to $f=0.002$, where $N=2\times {10}^4$. The synaptic release probability $p$ is calculated to yield the appropriate “effective network size” $pN$. Poisson-distributed random delays were used in computing the gain curve for $f=0.002$; the average delay was 6 ms on the bottom half and 4 ms on the top half of the gain curve. Note that, for large values of the external driving strength $f\nu$, the gain curves computed numerically appear to all asymptote toward the straight line given by Eq. (29), which is the asymptote of the gain curve (28) in the mean-driven limit for the same values of $\tau$ and $S$.

Figure 9

(Color online) Voltage probability density functions obtained via the Fokker-Planck equation (20) and numerical simulations for the external driving spike strength $f=0.002$ and the effective network size $pN=500$, for which Eq. (20) gives a valid approximation. Starting with the graph with the highest maximum and ending with the graph with the lowest maximum, the corresponding values of the external driving $f\nu$ are (a) 0.2, 0.22, 0.24, (b) 0.26, 0.32, 0.38, 0.44, and 0.5. Other parameter values are $\tau =1$, $S=0.1$, and ${\delta }_s=0.0002$. The reversal and threshold potentials satisfy Eq. (2). The minimal number of spike-induced voltage jumps needed for a neuron to reach $V_T$ beginning at ${\epsilon }_r$, computed from Eq. (14), is $\sim 100$. In the simulations, $N={10}^4$ and $p=0.05$. Poisson-distributed random delays were used in the simulation; the average delay was 4 ms. Inset (a): the locations of the displayed voltage probability density functions along the gain curve. The value of the probability density function’s maximum decreases with increasing value of $f\nu$ along the gain curve. Inset (b): the probability density functions obtained via simulations do not satisfy the boundary condition (10) due to finite rise and decay conductance times.

Figure 10

(Color online) Detail near $v=V_T$ of the voltage probability density computed via numerical simulations for a feed-forward neuron with the external driving spike strength $f=0.0025$ and the driving strength $f\nu =40$. Other parameter values are $\tau =1\left(=20\text{ms}\right)$, $S=0$, and $N=1$. The reversal and threshold potentials satisfy Eq. (2). The conductance rise time in Eq. (52) is instantaneous, ${\sigma }_r=0$. The decay conductance time $\sigma ={\sigma }_d$ is varied as shown in the legend, with $\delta \text{-PSP}$ denoting the instantaneous conductance time course used in Eq. (1). Note how the value of $\rho \left(V_T\right)$ decreases with decreasing ${\sigma }_d$ and vanishes for $\delta \text{-PSP}$.

Figure 11

(Color online) Voltage probability density (a) obtained theoretically from the Fokker-Planck equation (20) and (b) computed via numerical simulations for the fixed external driving spike strength $f=0.1$ and the effective network size $pN=10$, for which Eq. (20) is not a valid approximation. Other parameter values are $\tau =1$, $S=0.1$, and ${\delta }_s=0.0002$. The reversal and threshold potentials satisfy Eq. (2). The values of the external driving strength $f\nu$ increase from the graphs with the highest to the graphs with the lowest maxima. In the simulations, $N={10}^4$ and $p=0.001$. The minimal number of spike-induced voltage jumps needed for a neuron to reach $V_T$ beginning at ${\epsilon }_r$, computed from Eq. (14), is $\sim 2$.

Figure 12

(Color online) (a) Bistability and hysteresis in gain curves obtained theoretically from the Fokker-Planck equation (20) and the mean-driven approximation (28) and computed via numerical simulations for the external driving spike strength $f=0.002$ and the effective network size $pN=500$. Note that the mean-driven theory would predict the jump on the up-ramp branch of the gain curve obtained by the simulations to be initiated near the point $f\nu =B-1$, $m=0$. Other parameter values are $\tau =1$, $S=0.2$, and ${\delta }_s=0.0002$. The reversal and threshold potentials satisfy Eq. (2). In the simulations, $N={10}^4$ and $p=0.05$. For each value of $f\nu$, ${\tau }_{f\nu }=10\text{s}$ of the network evolution was simulated. The difference between two consecutive $f\nu$ values was $\Delta \left(f\nu \right)=0.001$, giving the speed ${10}^{-4}{\text{s}}^{-1}$. Poisson-distributed random transmission delays averaging 4 ms were used. (b) Hysteresis obtained in the simulations using fast and slow ramping speeds. The hysteretic gain-curve branches obtained for the slow ramping speed are the same as on the top. For the gain curves obtained by fast ramping, an ensemble of 50 simulation results is shown, with $N={10}^3$, $p=0.5$, ${\tau }_{f\nu }=1\text{s}$, $\Delta \left(f\nu \right)=0.0004$, and $\text{speed}=4\times {10}^{-4}{\text{s}}^{-1}$.