[Paper Review] Mechanism, dynamics, and biological existence of multistability in a large class of bursting neurons

Jonathan P. Newman, Robert J. Butera|arXiv (Cornell University)|Oct 8, 2008

stochastic dynamics and bifurcation24 references8 citations

TL;DR

This paper proposes that a large class of bursting neurons, particularly those with a circle/circle dynamical topology, inherently support multistability through saddle-node on invariant circle (SNIC) bifurcations, enabling stable coexistence of multiple bursting rhythms. Experimental evidence from *Aplysia californica* neurons shows sustained mode switches after perturbation, suggesting intrinsic, non-synaptic short-term memory via multirhythmic bursting dynamics.

ABSTRACT

Multistability, the coexistence of multiple attractors in a dynamical system, is explored in bursting nerve cells. A modeling study is performed to show that a large class of bursting systems, as defined by a shared topology when represented as dynamical systems, is inherently suited to support multistability. We derive the bifurcation structure and parametric trends leading to multistability in these systems. Evidence for the existence of multirhythmic behavior in neurons of the aquatic mollusc Aplysia californica that is consistent with our proposed mechanism is presented. Although these experimental results are preliminary, they indicate that single neurons may be capable of dynamically storing information for longer time scales than typically attributed to nonsynaptic mechanisms.

Motivation & Objective

• To identify the dynamical mechanism enabling multistability in a broad class of bursting neurons with shared topological structure.
• To explain the bifurcation structure and parametric conditions leading to multirhythmic bursting in these systems.
• To provide experimental evidence for multirhythmic behavior in identified neurons of *Aplysia californica*.
• To explore the biological plausibility of intrinsic, extra-synaptic information storage via stable switching between bursting modes.

Proposed method

• Modeling bursting neurons as singularly perturbed (SP) dynamical systems with fast (spiking) and slow (modulatory) subsystems.
• Reducing the system to a normal form with a quadratic integrate-and-fire fast subsystem and a damped linear oscillator slow subsystem.
• Using singular perturbation theory to analyze bifurcations, particularly the SNIC bifurcation, as the mechanism for switching between active and silent phases.
• Deriving inter-spike period and bursting dynamics using time-averaged perturbations from spike-induced resets of slow variables.
• Applying one-dimensional map analysis to study the stability and coexistence of multiple bursting rhythms.
• Conducting intracellular recordings from desheathed *Aplysia californica* abdominal ganglia under low-Ca²⁺, high-Mg²⁺ conditions to isolate intrinsic dynamics.

Experimental results

Research questions

• RQ1 What dynamical mechanism enables multistability in a large class of bursting neurons with a shared topological structure?
• RQ2 How do bifurcation parameters and system topology give rise to coexisting bursting rhythms in single neurons?
• RQ3 Can experimentally induced perturbations lead to sustained switching between distinct bursting modes in identified invertebrate neurons?
• RQ4 Is there biological evidence for multirhythmic bursting in *Aplysia californica* neurons consistent with the proposed model?
• RQ5 Can intrinsic, non-synaptic neuronal dynamics support stable, long-lasting information storage through mode switching?

Key findings

• A large class of bursting neurons with a circle/circle topology are inherently capable of multistability due to SNIC bifurcations in their fast subsystem.
• The model predicts that slow subsystem dynamics, perturbed periodically by spikes, can support multiple stable bursting rhythms depending on initial conditions.
• Experimental recordings from *Aplysia californica* neurons show sustained mode switches following current injection in four out of eleven cells, with three cells showing multiple switches.
• Cells exhibiting multirhythmicity maintained altered bursting patterns (e.g., spike count changes) for at least seven bursts post-perturbation, indicating stable attractor switching.
• The observed multirhythmic behavior is consistent with the theoretical model's prediction of coexisting attractors under fixed parameters.
• The results suggest that single neurons can act as non-synaptic short-term memory elements by storing sensory input as a stable change in intrinsic bursting pattern.