Cascade-induced synchrony in stochastically driven neuronal networks

Perfect spike-to-spike synchrony is studied in all-to-all coupled networks of identical excitatory, current-based, integrate-and-fire neurons with delta-impulse coupling currents and Poisson spike-train external drive. This synchrony is induced by repeated cascading “total firing events,” during which all neurons fire at once. In this regime, the network exhibits nearly periodic dynamics, switching between an effectively uncoupled state and a cascade-coupled total firing state. The probability of cascading total firing events occurring in the network is computed through a combinatorial analysis conditioned upon the random time when the first neuron fires and using the probability distribution of the subthreshold membrane potentials for the remaining neurons in the network. The probability distribution of the former is found from a first-passage-time problem described by a Fokker-Planck equation, which is solved analytically via an eigenfunction expansion. The latter is found using a central limit argument via a calculation of the cumulants of a single neuronal voltage. The influence of additional physiological effects that hinder or eliminate cascade-induced synchrony are also investigated. Conditions for the validity of the approximations made in the analytical derivations are discussed and verified via direct numerical simulations.

Figure 1

(Color online) Raster plots of firing times for the system of $N=1000$ neurons (only 100 shown) with initial voltages chosen randomly between $V_R$ and $V_T$. For the synchronizable network (c), with $f=0.0002$, $f\nu =1.2$ and $S=10.0$, one total firing event is followed by another total firing event with high probability, $P\left(C\right)=0.952$. The other two systems, (a) with $f=0.01$, $f\nu =1.2$ and $S=0.5$, and (b) with $f=0.02$, $f\nu =1.2$, and $S=1.0$, do not satisfy our stringent definition of synchronizability.

Figure 2

Average cascade size per firing event as a function of spike strength, $f$, and coupling strength, $S$, for a network of $N=100$ neurons with initial voltages chosen randomly between $V_R$ and $V_T$. (top) Subthreshold regime, $f\nu =0.9$ and (bottom) superthreshold regime, $f\nu =1.2$. If no firing events were detected during the $t=400$ run, the cascade size was assigned to be zero.

Figure 3

(Color online) The relation between the probability, $P\left(T>t\right)$ in Eq. (5), of a single neuron to have not yet fired (inset) and the solution, $p_v\left(x,t\right)$, to the Fokker-Planck equation [Eq. (7)] at the indicated points in time with $f=0.01$ and $f\nu =0.95$.

Figure 4

(Color online) The pdf, $p_T\left(t\right)$, for the first exit time of a single, uncoupled, neuron (Eq. (11), solid line, blue online) and the pdf, $p_T^{\left(1\right)}\left(t\right)$, for the first exit time of $N=500$ neurons [Eq. (3), dashed line, red online] are compared to the results from 2000 numerical simulations of the single neuron and a network of 500 neurons (triangles and circles, respectively). The values $f=0.001$ and $f\nu =1.0$ are used.

Figure 5

(Color online) Gaussian approximation for the pdf, $p_v\left(x,t\right)$ in Eq. (14a), of the voltage of a typical neuron (dark gray, blue online), and the pdf, $p_v^{\left(N\right)}\left(x,t\right)$ in Eq. (16), of the maximum voltage of a set of $N=500$ neurons (gray, green online) at time $t=1.5$, compared with results from Monte Carlo simulations of the network [Eq. (1)]. The values $f=0.001$ and $f\nu =1.0$ are used.

Figure 6

(Color online) The dependence of the solutions ${\mu }_N$ in Eq. (21) (dash-dotted line, red online) and $y_{\text{max}}\left(N\right)$ (dashed black line), along with its approximation in Eq. (24) (gray line, green online) on the network size, $N$. The approximation in Eq. (24) is derived assuming $N⪢1$.

Figure 7

(Color online) The firing rates determined by the methods in Sec. 4A (solid lines) and Sec. 4B (dashed lines) are compared with those obtained from numerical simulations (open circles) of the system [Eq. (1)] for 300 time units, as a function of $f\nu$. In (a) the gain curves for the indicated values of $f$ while $N=100$ and $S=10$ show the synchronized stochastic network fires at a rate faster than ${\widehat{\tau }}^{-1}$, the synchronized deterministic network firing rate in Eq. (2) (black line). In (b) the gain curves are plotted for the indicated values of $N$ while $f=0.01$ and $S=20$. The inset in (a) compares the results of simulations (symbols) to the first term in the expansion of $1/{\tau }_N-1/\widehat{\tau }$ about $f=0$, indicating a square root dependence of the firing rate, $m$, on the size of the fluctuations, $f$, for constant $f\nu =1.20$, $N=100$ and $S=10$. The inset in (b) compares the results of simulations (symbols) to $1/{\tau }_N-1/\widehat{\tau }$ computed with ${\mu }_N$ from Eq. (21) (solid line) and ${\mu }_N=y_{\text{max}}\left(N\right)$, using the approximation in Eq. (24) (dash-dot line), indicating a logarithmic dependence of the firing rate, $m$, on the size of the network, $N$, for constant $f\nu =1.2$, $f=0.01$, and $S=20$.

Figure 8

(Color online) Probability that condition (25) holds, $P\left(C\right)$, vs network coupling strength scaled by network size, $S/N$, computed with the method in Sec. 5A (lines) and from numerical Monte Carlo simulation of system (1) (symbols), averaged over 500 simulations. Top: for the indicated values of $f$, $N=100$ and $f\nu =1.2$. Bottom: for the indicated values of $N$, $f\nu =1.2$ and $f=0.001$. Inset: same data plotted vs $S$.

Figure 9

(Color online) Probability that condition (25) holds, $P\left(C\right)$, vs mean external current, $f\nu$, computed with the method in Sec. 5A (solid lines) and from Monte Carlo numerical simulation of system (1) (symbols), averaged over 500 samples. For each curve with the indicated values of $f$, $N=100$ and $S=1.5$. Inset: Value of $f\nu$ at the location of the transition [where $P\left(C\right)=0.5$] between asynchronous and synchronous behavior as a function of fluctuation size, $f$.

Figure 10

(Color online) Behavior of the network [Eq. (1)] with the addition of (top) synaptic failure and (bottom) sparse random network connections. Theoretical computation with modification to Eq. (30) (solid lines) is compared to results from numerical Monte Carlo simulations (circles with error bars), averaged over 500 simulations and (bottom) 10 networks. The values $f=0.001$, $f\nu =1.2$, $S=2.0$, and (right) the indicated values of $p_f$ and $p_c$, (top left) $p_f=0.90$, (bottom left) $p_c=0.90$ are used.

Figure 11

(Color online) Raster plots demonstrating the behavior of the network with random transmission delay. $N=100$, $f=0.001$, $f\nu =1.2$, $S=0.1$; (top) no delay (middle) average delay 0.002 (bottom) average delay 0.2

Figure 12

(Color online) Raster plots for the network with the delta functions in Eq. (1b) replaced by $t^2/{\tau }_E^2\text{exp}\left(-t/{\tau }_E\right)$. $N=1000$, $f=0.001$, $f\nu =1.2$, $S=0.1$; (top) ${\tau }_E=3\times {10}^{-6}$ (middle) ${\tau }_E=3\times {10}^{-2}$ (bottom) ${\tau }_E=3\times {10}^{-1}$

Figure 13

(Color online) Gain curve comparison between the theory (solid line) derived in Sec. 4A and full numerical simulations of the system [Eq. (1)] (symbols) when $P\left(C\right)<0.85$, our cutoff for classifying the network as synchronous. For the two values of internal coupling, we keep $N=100$ and $f=0.001$ constant. Inset: the value of $P\left(C\right)$ obtained from Monte Carlo numerical simulations (500 samples) for both values of $S$ as a function of $f\nu$.