Network-induced chaos in integrate-and-fire neuronal ensembles

It has been shown that a single standard linear integrate-and-fire (IF) neuron under a general time-dependent stimulus cannot possess chaotic dynamics despite the firing-reset discontinuity. Here we address the issue of whether conductance-based, pulsed-coupled network interactions can induce chaos in an IF neuronal ensemble. Using numerical methods, we demonstrate that all-to-all, homogeneously pulse-coupled IF neuronal networks can indeed give rise to chaotic dynamics under an external periodic current drive. We also provide a precise characterization of the largest Lyapunov exponent for these high dimensional nonsmooth dynamical systems. In addition, we present a stable and accurate numerical algorithm for evaluating the largest Lyapunov exponent, which can overcome difficulties encountered by traditional methods for these nonsmooth dynamical systems with degeneracy induced by, e.g., refractoriness of neurons.

Figure 1

(Color online) ${\lambda }_{\text{max}}$ obtained using different computational methods vs network coupling strength $S$. The thick dark solid line corresponds to the result generated by our algorithm, the thin dark gray dash line corresponds to the result generated by the standard algorithm and the thick light gray solid line (cyan online) corresponds to the result generated by Muller’s method.

Figure 2

(Color online) (a) ${\lambda }_{\text{max}}$ vs network coupling strength $S$ (b) last 80 ISI is plotted for each value of $S$. Inset: average firing frequency vs $S$. In a reduced-dimensional units (with $G$ being in the unit of $\left[{\text{ms}}^{-1}\right]$), parameters are chosen as $G^L=0.05{\text{ms}}^{-1}$, ${ϵ}^L=0$, ${ϵ}^E=14/3$, $V_T=1$, $V_R=0$, $\sigma =2\text{ms}$, ${\tau }_{\text{ref}}=2\text{ms}$, $I_0=0.05{\text{ms}}^{-1}$, $I_1=0.05{\text{ms}}^{-1}$, $\mu =0.04{\text{ms}}^{-1}$ [53, 54], which correspond to typical physiological values: $G^L=50\times {10}^{-6}{\Omega }^{-1}{\text{cm}}^{-2}$, ${ϵ}^L=-70\text{mV}$, ${ϵ}^E=0\text{mV}$, $V_T=-55\text{mV}$.

Figure 3

(Color online) Convergence of numerical computation of ${\lambda }_{\text{max}}$ vs network coupling strength $S$ over different time steps $\Delta t=1/16\text{ms}$ (dark gray: red online), 1/128 ms (light gray: cyan online), 1/1024 ms (dark: black online).

Figure 4

(Color online) (a1)–(c1) Return map of the firing phase in periodic (a1), chaotic (b1), and quasiperiodic (c1) state; (a2)–(c2) power spectra, averaged over all neurons, of membrane potential traces in the network in periodic (a2), chaotic (b2), and quasiperiodic (c2) state; (a3)–(c3) convergence of the modified fourth order Runge-Kutta method in periodic (a3), chaotic (b3), and quasiperiodic (c3) state. The triangle (linked by thin dark gray dash line: black online) corresponds to the relative average error in the voltage, the square (linked by thick light gray solid line: cyan online) corresponds to the relative error in the average voltage and the thick dark straight line corresponds to a scaling with exponent 4 between the error and the time step.