Stability of the splay state in pulse-coupled networks

The stability of the dynamical states characterized by a uniform firing rate (splay states) is analyzed in a network of globally coupled leaky integrate-and-fire neurons. This is done by reducing the set of differential equations to a map that is investigated in the limit of large network size. We show that the stability of the splay state depends crucially on the ratio between the pulse width and the interspike interval. More precisely, the spectrum of Floquet exponents turns out to consist of three components: (i) one that coincides with the predictions of the mean-field analysis [Abbott and van Vreesvijk, Phys. Rev. E 48, 1483 (1993)], (ii) a component measuring the instability of “finite-frequency” modes, (iii) a number of “isolated” eigenvalues that are connected to the characteristics of the single pulse and may give rise to strong instabilities (the Floquet exponent being proportional to the network size). Finally, as a side result, we find that the splay state can be stable even for inhibitory coupling.

Figure 1

(Color online) Unitary circle and exact Floquet multipliers spectra for the complete maps for $N=20$ (red circles) and $N=10$ (blue crosses) for excitatory coupling. The parameters are $a=3.0$, $g=0.4$, and $\alpha =30.0$.

Figure 2

(Color online) Phase diagram for the stability of the splay state in a neural network with excitatory coupling acting through finite pulse width. The solid line separating the stable from the unstable regions in the $(g,\alpha )$ plane has been derived from the analytic formula of the Floquet spectrum (22) with $a=1.3$. Please notice that in this context “stable” refers to finite $N$, for infinite systems the stability will become marginal.

Figure 3

(Color online) (a) Log-log plot of the absolute values of the Floquet exponents ${\lambda }_k$, ordered from the largest to the smallest as a function of the index $k=1,\dots ,N$ for $N=100,200,400$. The dashed line has slope $-2$. (b) The Floquet exponent as a function of the rescaled phase $\phi N$, for $N=100$ (black circles) and $N=200$ (red crosses). In both pictures the parameter values are $a=3.0$, $g=0.4$, and $\alpha =30.0$.

Figure 4

(Color online) Floquet exponents ${\lambda }_k\left(\phi \right)$ as a function of the phase ${\phi }_k$ for finite pulse width $\alpha =30.0$ and finite neuron numbers $N=500$ in the case of excitatory coupling $g=0.4$. The filled circles represent the exact result for finite $N$, while the red empty squares and the blue empty triangles refer to approximated results correct up to the first or second order in $1∕N$, respectively. The parameters here are $a=3$.

Figure 5

(Color online) Floquet spectrum $\lambda \left(\phi \right)$ as a function of the phase $\phi$ in the case of vanishing pulse width. Red circles indicate the numerical results obtained for the linear stability analysis without any approximation for $N=1000$, the black line refers to the first-order approximate expression (32), and the dashed blue line to the approximation (30). In the inset we magnify the region of small values of $\phi$. The parameters are $a=1.3$, $g=-1.2$, and $\beta =1.0$.

Figure 6

(Color online) Maximal Floquet exponent ${\lambda }_m$ (red diamonds and dashed line) and second Floquet exponent (blue filled circles) as a function of $\beta$ for $a=1.3$ and $g=-1.2$. The exponents have been numerically obtained by diagonalizing the matrix that corresponds to $N=500$ neurons. The black solid line corresponds to the analytical expression (33) for ${\lambda }_{\pi }$. In the inset, the rescaled Floquet exponent ${\lambda }_m∕N$ (red diamonds) is compared to the analytical expression (34) for ${\lambda }_{N,N+1}$ (black solid line).

Figure 7

(Color online) Finite-frequency Floquet spectra $\lambda \left(\phi \right)$ for $\beta =0.01$ (black solid line), 0.02 (red dashed line), 0.04 (green dot-dashed line), and 0.08 (blue dotted line) and $a=1.3$, $g=-1.2$.

Figure 8

(Color online) Phase diagram for the stability of the splay state for $a=1.3$ and inhibitory coupling in the sharp spike limit. The solid black horizontal line divides the stable from the unstable region, while the dashed red line encircles the strongly unstable phase where the maximum Floquet exponent is proportional to $N$.

Figure 9

(Color online) Time evolution of the field $E\left(t\right)$ associated to a splay state for increasing $N$ values: $N=100$ (black solid lines), 200 (blue dashed lines), 300 (red dotted lines). The upper panel refers to a fixed value of $\alpha =120$, while the lower panel to a constant $\beta =0.14$. The other parameters are $a=1.3$ and $g=-1.2$