Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map

LoFaro, Thomas; Kopell, Nancy

doi:10.1007/s002850050157

Cited by 21 publications

(7 citation statements)

References 31 publications

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“…Several studies of neuronal systems have used reduction to low-dimensional maps to prove the existence and stability of solutions [6, 15, 14, 4, 19, 18], most by tracking state variables in a low-dimensional phase space. The analysis on the slow manifold in our study made it possible to derive a one-dimensional map Π of the unit interval whose dynamics predicted the behavior of the full set of differential equations (1).…”

Section: Discussionmentioning

confidence: 99%

The Influence of the A-Current on the Dynamics of an Oscillator-Follower Inhibitory Network

Bose¹,

Nadim²

2009

SIAM J. Appl. Dyn. Syst.

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The transient potassium A-current is present in almost all neurons and plays an essential role in determining the timing and frequency of action potential generation. We use a three-variable mathematical model to examine the role of the A-current in a rhythmic inhibitory network, as is common in central pattern generation. We focus on a feed-forward architecture consisting of an oscillator neuron inhibiting a follower neuron. We use separation of time scales to demonstrate that the trajectory of the follower neuron within each cycle can be tracked by analyzing the dynamics on a 2-dimensional slow manifold that as determined by the two slow model variables: the recovery variable and the inactivation of the A-current. The steady-state trajectory, however, requires tracking the slow variables across multiple cycles. We show that tracking the slow variables, under simplifying assumptions, leads to a one-dimensional map of the unit interval with at most a single discontinuity depending on gA, the maximal conductance of the A-current, or other model parameters. We demonstrate that, as the value of gA is varied, the trajectory of the follower neuron goes through a set of bifurcations to produce n:m periodic solutions where the follower neuron becomes active m times for each n cycles of the oscillator. Using a generalized Pascal triangle, each n:m trajectory can be constructed as a combination of solutions from a higher level of the triangle.

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Section: Discussionmentioning

confidence: 99%

The Influence of the A-Current on the Dynamics of an Oscillator-Follower Inhibitory Network

Bose¹,

Nadim²

2009

SIAM J. Appl. Dyn. Syst.

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“…In some previous studies see 14,24 , the slow manifolds were one dimensional, and one could naturally de ne a metric between the two cells. The metric could either beà time-metric' which measures the time it takes for the`trailing cell' to reach the position of the`leading cell' or a`space-metric' which measures the Euclidean distance between the two cells on the slow manifold.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of two mutually coupled slow inhibitory neurons

Terman

Kopell

Bose

1998

Physica D: Nonlinear Phenomena

153

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Inhibition in oscillatory networks of neurons can have apparently paradoxical e ects, sometimes creating dispersion of phases, sometimes fostering synchrony in the network. We analyze a pair of biophysically modeled neurons and show h o w the rates of onset and decay of inhibition interact with the time scales of the intrinsic oscillators to determine when stable synchrony is possible. We show that there are two di erent regimes in parameter space in which di erent combinations of the time constants and other parameters regulate whether the synchronous state is stable. We also discuss the construction and stability of non-synchronous solutions, and the implications of the analysis for larger networks. The analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow ows and fast jumps.

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“…Moreover, their analysis was restricted to periodic forcing, whereas ours accommodates, and indeed is particularly well suited for, stochasticity in input timing. LoFaro and Kopell (1999) also used 1D maps to study a forced excitable system, but in their work, the excitable system was a neuron mutually coupled via inhibitory synapses to an oscillatory cell and the map was a singular Poincaré map, with each iteration corresponding to the time between jumps to the active phase. Similarly, Keener, Hoppensteadt, and Rinzel (1981) and subsequent authors have used firing time maps to study mode locking in integrateand-fire and related models with periodic stimuli.…”

Section: Discussionmentioning

confidence: 99%

The Firing of an Excitable Neuron in the Presence of Stochastic Trains of Strong Synaptic Inputs

Rubin

Josić

2007

Neural Computation

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We consider a fast-slow excitable system subject to a stochastic excitatory input train and show that under general conditions, its long-term behavior is captured by an irreducible Markov chain with a limiting distribution. This limiting distribution allows for the analytical calculation of the system's probability of firing in response to each input, the expected number of response failures between firings, and the distribution of slow variable values between firings. Moreover, using this approach, it is possible to understand why the system will not have a stationary distribution and why Monte Carlo simulations do not converge under certain conditions. The analytical calculations involved can be performed whenever the distribution of interexcitation intervals and the recovery dynamics of the slow variable are known. The method can be extended to other models that feature a single variable that builds up to a threshold where an instantaneous spike and reset occur. We also discuss how the Markov chain analysis generalizes to any pair of input trains, excitatory or inhibitory and synaptic or not, such that the frequencies of the two trains are sufficiently different from each other. We illustrate this analysis on a model thalamocortical (TC) cell subject to two example distributions of excitatory synaptic inputs in the cases of constant and rhythmic inhibition. The analysis shows a drastic drop in the likelihood of firing just after inhibitory onset in the case of rhythmic inhibition, relative even to the case of elevated but constant inhibition. This observation provides support for a possible mechanism for the induction of motor symptoms in Parkinson's disease and for their relief by deep brain stimulation, analyzed in Rubin and Terman (2004).

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Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map

Cited by 21 publications

References 31 publications

The Influence of the A-Current on the Dynamics of an Oscillator-Follower Inhibitory Network

The Influence of the A-Current on the Dynamics of an Oscillator-Follower Inhibitory Network

Dynamics of two mutually coupled slow inhibitory neurons

The Firing of an Excitable Neuron in the Presence of Stochastic Trains of Strong Synaptic Inputs

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