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Neural Dynamics
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The FitzHugh-Nagumo v-Nullcline (Iap=0)
W=0 v -1 1 0<W<1 v V derivative is 0 at the hash markes Connect the hash marks for all w to form the v-nullcline
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Morris-Lecar Phase Plane
Red: subthreshold response, Green: impulse
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Equilibria
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Example of Linearization Function of 1 Variable
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Example of Linearization Function of 2 Variables
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Effect of Multiplying a Vector by a Matrix
Ax Typical effect of matrix-vector multiplication Multiplication by eigenvectors Red: 𝜆>1 Blue: 𝜆<−1
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Tonic Spiking in Morris-Lecar Model
With enough applied current the model produces tonic or periodic spiking
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Recording of Tonic Spiking
Tonic response Single response Spiking response to injected current recorded from a midbrain sensory neuron. From Kohashi and Carlson, 2014.
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Examples of Depolarization Block
Two examples of depolarization block in midbrain dopamine neurons. From Qian et al., 2014.
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Bistability Between Steady State and Limit Cycle
An unstable limit cycle forms the separatrix
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A Bistable System 3 1 Potential energy function
Noise can kick the system from one basin into the other
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Morris-Lecar Bifurcation Analysis
A type-2 neural oscillator. Black: stationary branch, Red: periodic branch
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Three Examples of Hodgkin’s Three Classes of Neurons
Hodgkin (1948) classified neurons into three types based on their response to depolarizing current pulses Recordings from spinal lamina 1 neurons. From Prescott, De Koninck, and Sejnowski, PLoS Comput. Biol., 2008
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Type 2 Dynamics with Morris-Lecar
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A Type-1 Neural Oscillator
Oscillation period approaches infinity near the homoclinic bifurcation
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Example of a Saddle Point in a Planar System
Unstable manifold Stable manifold The saddle point is at the origin in this example.
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Example of a Saddle Point in Terms of Potential Energy
Stable manifold
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Sub- and Super-threshold Responses
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In Morris-Lecar the Threshold is the Middle Branch
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Integrate-and-Fire Model Neuron
The spike is drawn by hand
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Integrate-and-Fire with Adaptation
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