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On the local stability of the elapsed-time model in terms of the transmission delay and interconnection strength

María J. Cáceres, José A Cañizo, Nicolas Torres

TL;DR

This work analyzes the local (linear) stability of the nonlinear elapsed-time model for interconnected neurons with a discrete synaptic delay. By reformulating the linearized dynamics as Volterra-type integral equations and applying Laplace transform techniques, the authors derive a pole-based stability criterion governed by the delay $d$ and the state-dependent connectivity $A^*$. The central object is the delay function $\Phi_d(z)=1-e^{-zd}(\widehat{h_0}(z)+A^*)$, whose zeros determine stability: all zeros with negative real part imply asymptotic stability, while any zero with positive real part implies instability. They further specialize to an absolute refractory period to obtain explicit thresholds, provide a rigorous asymptotic framework for the modified Volterra equation with delay, and illustrate the results with numerical bifurcation diagrams showing how delay and interconnection strength shape the long-time behavior, including Hopf-like transitions and possible periodic regimes. The study highlights $A^*$ as a key quantitative measure of interconnection strength and demonstrates how delay can destabilize or destabilize-and-regulate the system, offering a path toward understanding complex neuronal population dynamics in delayed, nonlinear settings.

Abstract

The elapsed-time model describes the behavior of interconnected neurons through the time since their last spike. It is an age-structured non-linear equation in which age corresponds to the elapsed time since the last discharge, and models many interesting dynamics depending on the type of interactions between neurons. We investigate the linearized stability of this equation by considering a discrete delay, which accounts for the possibility of a synaptic delay due to the time needed to transmit a nerve impulse from one neuron to the rest of the ensemble. We state a stability criterion that allows to determine if a steady state is linearly stable or unstable depending on the delay and the interaction between neurons. Our approach relies on the study of the asymptotic behavior of related Volterra-type integral equations in terms of theirs Laplace transforms. The analysis is complemented with numerical simulations illustrating the change of stability of a steady state in terms of the delay and the intensity of interconnections.

On the local stability of the elapsed-time model in terms of the transmission delay and interconnection strength

TL;DR

This work analyzes the local (linear) stability of the nonlinear elapsed-time model for interconnected neurons with a discrete synaptic delay. By reformulating the linearized dynamics as Volterra-type integral equations and applying Laplace transform techniques, the authors derive a pole-based stability criterion governed by the delay and the state-dependent connectivity . The central object is the delay function , whose zeros determine stability: all zeros with negative real part imply asymptotic stability, while any zero with positive real part implies instability. They further specialize to an absolute refractory period to obtain explicit thresholds, provide a rigorous asymptotic framework for the modified Volterra equation with delay, and illustrate the results with numerical bifurcation diagrams showing how delay and interconnection strength shape the long-time behavior, including Hopf-like transitions and possible periodic regimes. The study highlights as a key quantitative measure of interconnection strength and demonstrates how delay can destabilize or destabilize-and-regulate the system, offering a path toward understanding complex neuronal population dynamics in delayed, nonlinear settings.

Abstract

The elapsed-time model describes the behavior of interconnected neurons through the time since their last spike. It is an age-structured non-linear equation in which age corresponds to the elapsed time since the last discharge, and models many interesting dynamics depending on the type of interactions between neurons. We investigate the linearized stability of this equation by considering a discrete delay, which accounts for the possibility of a synaptic delay due to the time needed to transmit a nerve impulse from one neuron to the rest of the ensemble. We state a stability criterion that allows to determine if a steady state is linearly stable or unstable depending on the delay and the interaction between neurons. Our approach relies on the study of the asymptotic behavior of related Volterra-type integral equations in terms of theirs Laplace transforms. The analysis is complemented with numerical simulations illustrating the change of stability of a steady state in terms of the delay and the intensity of interconnections.

Paper Structure

This paper contains 12 sections, 12 theorems, 115 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results of this article
  3. The linear equation revisited
  4. Linear stability of the nonlinear system \ref{['eqdelay']}
  5. Eigenfunctions of the linearized system
  6. Reformulation as a modified Volterra integral equation
  7. Application: System with an absolute refractory period
  8. Asymptotic behavior for convolution Volterra equation with discrete delay
  9. Numerical simulations
  10. Bifurcation diagrams
  11. Stability for systems with $|A^*|<1$
  12. Conclusions and perspectives

Key Result

Theorem 1

Assume that $S$ satisfies boundS and consider the unique positive steady state $(n^*,r^*)$ such that $n^*\in L^1(0,\infty)$ with $\int_0^\infty n^*\,da=1$ given in steady-state-linear. Then there exist constants $C,\mu>0$ such that for all initial condition $n^0\in \mathcal{M}(\mathds{R}^+)$ the fol and with $\langle n^0\rangle\coloneqq \int_0^\infty n^0 da$.

Figures (7)

  • Figure 1: Bifurcation diagram for the activity $r^*$ of a stationary solution of \ref{['eqdelay']} with $S$ given by \ref{['S_b1']} in terms of the parameter $b$ with $d=0$.
  • Figure 2: Bifurcation diagram for the activity $r^*$ of a stationary solution of \ref{['eqdelay']} with $S$ given by \ref{['S_b2']} in terms of the parameter $b$ with $d=0$.
  • Figure 3: Real part of the dominant eigenvalue $z_0$ in terms of the delay $d$ for the firing coefficient $S_{0.43^2}$ given in \ref{['S_b2']}. In the axis for $d$, green indicates the values of delay where the equilibrium is linearly stable, while red means instability. The red points indicate the critical values for $d$ where $\Re(z_0)=0$.
  • Figure 4: Evolution on time of the activity $r(t)$ of the nonlinear system \ref{['eqdelay']} with firing coefficient $S_{0.43^2}$ given in \ref{['S_b2']} and considering the initial data \ref{['init-data1']} and the transmission delay $d=0.01$.
  • Figure 5: Evolution on time of the activity $r(t)$ of the nonlinear system \ref{['eqdelay']} with firing coefficient $S_{0.43²}$ given in \ref{['S_b2']} and considering the initial data \ref{['init-data2']} and the transmission delay $d=0.01$.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Theorem 1: Exponential convergence to the steady state of the linear system
  • Remark 1
  • Theorem 2
  • Theorem 3: Linear stability criterion
  • Theorem 4
  • proof : Proof of Theorem \ref{['doeblin-conv']}
  • Remark 2
  • Lemma 1
  • proof
  • Theorem 5
  • ...and 17 more