On the asymptotic behavior of the NNLIF neuron model for general connectivity strength

Abstract

We prove new results on the asymptotic behavior of the nonlinear integrate-and-fire neuron model. Among them, we give a criterion for the linearized stability or instability of equilibria, without restriction on the connectivity parameter, which provides a proof of stability or instability in some cases. In all cases, this criterion can be checked numerically, allowing us to give a full picture of the stable and unstable equilibria depending on the connectivity parameter and transmission delay. We also give further spectral results on the associated linear equation, and use them to give improved results on the nonlinear stability of equilibria for weak connectivity, and on the link between linearized and nonlinear stability.

Figures (4)

• Figure 1: Function $\boldsymbol{\frac{1}{I(N)}}$ (see \ref{['eq:I']}) for different values of the connectivity parameter $\boldsymbol{b}$. Each crossing with the diagonal corresponds to an equilibrium of equation \ref{['eq:NNLIF']}. Reset and firing potential are set to $V_R=1$ and $V_F=2$ respectively.
• Figure 2: Values of $\boldsymbol{N}$ solving $\boldsymbol{N(I(N))=1,}$ for each $\boldsymbol{b}$. Point $b=1$ corresponds to $b=V_F-V_R$ for the selected parameter values. Therefore, there is only one equilibrium for $b<1$, two equilibria for $1 < b < b_e$ with $b_e$ a certain critical value, and none for $b > b_e$. The dashed line has a vertical asymptote at $b=1$. Reset and firing potential are set to $V_R=1$ and $V_F=2$ respectively.
• Figure 3: Value of $S(b) := - b \int_{0}^{\infty} N_q(t) \,\mathrm{d} t = \frac{\mathrm{d}}{\mathrm{d}N} \vert_{N = N_\infty} \left( \frac{1}{I(N)} \right)$ for different values of $b$. Vertical lines at $b=-9.4$ and $b=1$ separate regions with different behaviours in terms of stability for equilibria of equation \ref{['eq:linearized']}: for $b<-9.4$ (one equilibrium) we find that $S(b) < -1$, which coincides with point 3 of Theorem \ref{['thm:stability-main']}, i.e. the region of $b$ where equilibria stability depends on the delay value; in the case $-9.4<b<1$ (one equilibrium), we get $|S(b)| = |b| \int_0^\infty |N_q(t)| \,\mathrm{d} t < 1$ (since the function $N_q(t)$ is numerically seen to be always monotone), leading us to the scenario in point 2 of Theorem \ref{['thm:stability-main']}, where regardless of the delay value, the equilibrium is stable. For any $b>1$ there are two equilibria: for the lower equilibrium (solid line) the situation is the same as in the previous case, while for the higher equilibrium (dashed line), $S(b) > 1$ and point 1 of Theorem \ref{['thm:stability-main']} is fulfilled. Reset and firing potential are set to $V_R=1$ and $V_F=2$ respectively.
• Figure 4: Stability map of the linearized equation. For each pair of values of the connectivity parameter and transmission delay $(b,d)$, this color map shows the stability (dark) or instability (white) for the solutions of the linearized equation \ref{['eq:linearized-pde']}. The vertical black solid line placed at $b_p \approx-9.4$ separates the region of unconditional stability (for any delay $d$), from that where stability depends on the value of $d$. The vertical dashed line at $b=1$ separates the region with a unique steady state (left) from the one with two steady states (right). For cases with two steady states (that is, to the right of the vertical dashed line) we consider the lower one. The stability condition considered to make this map is based on Theorems \ref{['thm:N-asymptotic-behavior-main']} and \ref{['thm:stability-main']}: for each $b$ we find its related equilibrium $p_\infty$, choosing the lower one if there are two equilibria; we differentiate $p_\infty$ and use it as initial condition for the linear system given by equation \ref{['eq:NNLIF-PDE']}, thus obtaining $q(v,t) = e^{tL_\infty}\partial_v p_\infty$ and $N_q(t)$. Cases 1 and 2 of Theorem \ref{['thm:stability-main']} account for all cases with $b > b_p$. In order to tell apart the cases with $b < b_p$, for each $d>0$ we compute $Q(k):=-b e^{-ikd}\int_{0}^{\infty}N_q(t)e^{ikt} \,\mathrm{d} t$, with $k\in\mathbb{R}$; finally we count the number of times that $Q(k)$ crosses the line $(1,\infty)$ taking into account its orientation. If the net balance of crossings is different from $0$, the equilibrium $p_\infty$ related to $(b,d)$ is unstable (see the proof of Theorem \ref{['thm:stability-main']} in Section \ref{['sec:linearized-sg']} for a better understanding of this). Otherwise we need to compute the zeros of function $\Phi_d(\xi) := 1+b\hat{N_q}(\xi) \exp(-\xi d)$, given in Theorem \ref{['thm:N-asymptotic-behavior-main']} in order to check whether the equilibrium is stable or not. Reset and firing potential are set to $V_R=1$ and $V_F=2$ respectively.

Theorems & Definitions (50)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• proof
• Theorem 1.4
• Lemma 2.1: Well-posedness estimate
• Lemma 2.2: Regularization estimate
• Proposition 2.3: Spectral gap estimate
• Remark 2.4
• proof