NeuronsPart4
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This document discusses Jennifer Houser's work modeling voltage-gated ion channels, spiking neurons, delay differential equations, and integrate-and-fire models. It provides an overview of these topics and describes Houser's goals of developing a mathematical model to represent feed-forward inhibition circuits and incorporating latency. Specifically, it discusses modeling ion channel gating variables, the Izhikevich spiking neuron model, delayed differential equations requiring initial values, extensions of the leaky integrate-and-fire model, and Houser's short and long-term goals of further investigating mechanisms and writing a paper.
![Delayed Differential Equations (DDE)
dx
dt
= f(x(t), x(t − τ))
Rate of change of event at some current time is dependent on the
behavior of the function at some previous
Models various time processes or stage events such as:
Maturation of a species from newborn to adolescent
Delayed birth rate
State variable x must correspond to the interval [t − τ, t], τ > 0
The equation requires initial values for x(t0) and x(t0 − τ)
MATLAB built-in solver for DDE
dde23
J. E. Forde. (2005) Delay differential equation models in mathematical biology (Unpublished doctoral dissertation). The University
Jennifer Houser Mathematical Biosciences Institute 2014 REU](https://image.slidesharecdn.com/74fa85d5-d56b-4328-a8f3-1e991839c0ca-150620194402-lva1-app6892/85/NeuronsPart4-7-320.jpg)
NeuronsPart4
- 1. Latency of Thalamocortical Fast-Spiking Interneurons in Schizophrenia Jennifer Houser East Tennessee State University July 24, 2014 Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 2. Overview Topics MATLAB Modeling Voltage-Gated Ion Channels Spiking Neurons Delay Differential Equations Integrate-and-Fire Model Future Work Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 3. Voltage-Gated Ion Channels Models ionic channel gating variables Recall that the conductance of an ionic channel is equal to the probability that the channel is open Rate at which a channel is open is related to the difference between the probability of a closed channel being opened and the probability of an open channel being closed dw dt = (1 − w)k1 − wk2 = k1 − (k1 + k2)w, for w = n, m, h Input arguments: membrane potential and current probability that the ionic channel is open Simplistic but requires numerical data as input P. Wallisch, M. E. Lusignan, M. D. Benayoun, T. I. Baker, A. S. Dickey, and N. G. Hatsopoulos, MATLAB for Neuroscientists. Academic Press, London, 2014. Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 4. Spiking Neuron Model System of two ordinary-differential equations and a reset condition developed by E. Izhikevich dv dt = 0.04v2 + 5v + 140 − u + I du dt = a(bv − u) If v ≥ 30, then c → v and u + d → u v: membrane potential u: membrane recovery variable Accounts for activation of K+ channels and inactivation of Na+ channels E. M. Izhikevich. Simple model of spiking neurons. IEEE Transactions on Neural Networks. 14: 1569-1572, 2003. Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 5. Regular Spiking Neuron Plot P. Wallisch, M. E. Lusignan, M. D. Benayoun, T. I. Baker, A. S. Dickey, and N. G. Hatsopoulos, MATLAB for Neuroscientists. Academic Press,Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 6. Fast Spiking Neuron Plot P. Wallisch, M. E. Lusignan, M. D. Benayoun, T. I. Baker, A. S. Dickey, and N. G. Hatsopoulos, MATLAB for Neuroscientists. Academic Press,Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 7. Delayed Differential Equations (DDE) dx dt = f(x(t), x(t − τ)) Rate of change of event at some current time is dependent on the behavior of the function at some previous Models various time processes or stage events such as: Maturation of a species from newborn to adolescent Delayed birth rate State variable x must correspond to the interval [t − τ, t], τ > 0 The equation requires initial values for x(t0) and x(t0 − τ) MATLAB built-in solver for DDE dde23 J. E. Forde. (2005) Delay differential equation models in mathematical biology (Unpublished doctoral dissertation). The University Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 8. Integrate-and-Fire Models Does not take into account the underlying mechanisms of action potentials Utilizes a threshold value for the membrane potential that triggers an action potential vth Membrane potential is reset to subthreshold level after the occurrence of an action potential vr Ability to incorporate refractory period tref Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 9. Leaky Integrate-and-Fire (LIF) Model τm dv dt = −v(t) + RI(t) v(t): membrane potential as a function of time τm: membrane time constant R: resistance I(t): current as a function of time Simplest form of LIF model Easy to compute numerically and analytically Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 10. Extensions of LIF Model Include a refractory period involving a delay in the action potential once the membrane potential has been reset Constant input current: I(t) = I Models periodic behavior of action potentials Time-varying input current: I(t) Describes the membrane potential between action potential occurrences Synaptic currents Describes the spiking behavior due to pre-synaptic currents Considers the summation of individual pre-synaptic events M. Beierlein, J. R. Gibson, and B. W. Connors. ”Two dynamically distinct inhibitory networks in layer 4 of the neocortex.” J. Neurophsiol., 90: 2987 - 3000, 2003. Jennifer Houser Mathematical Biosciences Institute 2014 REU
- 11. Future Work Short-Term Goals Study integrate-and-fire model Further investigate the mechanisms of the feed-forward inhibition circuit Develop mathematical model to represent feed-forward inhibition Learn how to incorporate latency into model Long-Term Goals Write model section of paper Incorporate a latency term in model Jennifer Houser Mathematical Biosciences Institute 2014 REU