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Geometric Stability Switch Criteria In Delay Differential Systems With Delay Dependent Parameters
 SIAM. J. Math. Anal
, 2002
"... In most applications of delay di#erential equations in population dynamics,the need of incorporation of time delays is often the result of the existence of some stage structure. Since the throughstage survival rate is often a function of time delays,it is easy to conceive that these models may invo ..."
Abstract

Cited by 58 (6 self)
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In most applications of delay di#erential equations in population dynamics,the need of incorporation of time delays is often the result of the existence of some stage structure. Since the throughstage survival rate is often a function of time delays,it is easy to conceive that these models may involve some delay dependent parameters. The presence of such parameters often greatly complicates the task of an analytical study of such models. The main objective of this paper is to provide practical guidelines that combine graphical information with analytical work to e#ectively study the local stability of some models involving delay dependent parameters. Specifically,we shall show that the stability of a given steady state is simply determined by the graphs of some functions of # which can be expressed explicitly and thus can be easily depicted by Maple and other popular software. In fact,for most application problems,we need only look at one such function and locate its zeros. This function often has only two zeros,providing thresholds for stability switches. The common scenario is that as time delay increases,stability changes from stable to unstable to stable, implying that a large delay can be stabilizing. This scenario often contradicts the one provided by similar models with only delay independent parameters. Ke words. delay di#erential equations,stability switch,characteristic equations,stage structure, population models AMS sub je classifications. 34K18,34K20,92D25 PII. S0036141000376086 1.
Dynamics of Membrane Excitability Determine Interspike Interval Variability: A Link Between Spike Generation Mechanisms and Cortical Spike Train Statistics
, 1998
"... We propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains. The key lies in the nonlinear dynamics of cortical spike generation, which are consistent with type I membranes where saddlenode dynamics underlie excitability (Rinzel & Ermentrout ..."
Abstract

Cited by 52 (5 self)
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We propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains. The key lies in the nonlinear dynamics of cortical spike generation, which are consistent with type I membranes where saddlenode dynamics underlie excitability (Rinzel & Ermentrout, 1989). We present a canonical model for type I membranes, the θneuron. The θneuron is a phase model whose dynamics reflect salient features of type I membranes. This model generates spike trains with coefficient of variation (CV) above 0.6 when brought to firing by noisy inputs. This happens because the timing of spikes for a type I excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses. A noisy input current, giving random amplitude “kicks” to the cell, evokes highly irregular firing across a wide range of firing rates; an intrinsically oscillating cell gives regular spike trains. We corroborate the results with simulations of the MorrisLecar (ML) neural model with random synaptic inputs: type I ML yields high CVs. When this model is modified to have type II dynamics (periodicity arises via a Hopf bifurcation), however, it gives regular spike trains (CV below 0.3). Our results suggest that the high CV values such as those observed in cortical spike trains are an intrinsic characteristic of type I membranes driven to firing by “random” inputs. In contrast, neural oscillators or neurons exhibiting type II excitability should produce regular spike trains.