Mechanisms of action potential generation:

2D models

Action potential generation

Frequency-current relationships of neurons

Cortical pyramidal cells Cortical interneurons

Action potential generation in the HH model

CdV

dt= −gL(V − VL)− gNa(V − VNa)m3h− gK(V − VK)n4 + Iapp

τm(V )dm

dt= −m+m∞(V ) (fast positive feedback on voltage)

τh(V )dh

dt= −h+ h∞(V ) (slower negative feedback on voltage)

τn(V )dn

dt= −n+ n∞(V ) (slower negative feedback on voltage)

Summary

• Hodgkin-Huxley formalism very successful in terms of describing quantitatively action

potential generation;

• The model is quite complex mathematically - can we capture single neuron behavior

with simpler models?

• Reduction to a 2D model: Krinsky-Kokoz;

• Related 2D models: Morris Lecar, FitzHugh Nagumo;

• Analyzing 2D models: phase plane, stability analysis, bifurcations

• Bifurcations leading to periodic firing

Reduction to a 2-D model (Krinsky-Kokoz-Rinzel)

• Variablem is much faster than all other

variables:

⇒ m(t) = m∞(V )

• Dynamics of n and 1− h are similar:

⇒ h = 1− n

• Gives a 2-D model:

CdV

dt= −gL(V − VL)− gNa(V − VNa)m3

∞(V )(1− n)− gKn4(V − VK)

τn(V )dn

dt= −n+ n∞(V )

2D models

• Krinsky-Kokoz (reduction from HH)

CdV

dt= −gL(V − VL)− gNa(V − VNa)m3

∞(V )(1− n)− gKn4(V − VK)

+Iapp

τn(V )dn

dt= −n+ n∞(V )

• Morris-Lecar (giant muscle fiber of the barnacle)

CdV

dt= −gL(V − VL)− gCa(V − VCa)m∞(V )− gKn(V − VK) + Iapp

τn(V )dn

dt= −n+ n∞(V )

• FitzHugh-Nagumo (simpler model of excitability)

dV

dt= V (V − a)(1− V )− w + Iapp

dw

dt= ε(V − γw)

Analyzing 2D models - phase plane, nullclines

dV

dt= f(V, n)

dn

dt= g(V, n)

• V -nullcline: f(V, n) = 0

• n-nullcline: g(V, n) = 0

• Fixed points: intersections of both null-

clines

Fixed points and their stability

dV

dt= f(V, n)

dn

dt= g(V, n)

• Fixed points V∗, n∗ such that f(V∗, n∗) = g(V∗, n∗) = 0

• Small perturbation around fixed point,

V = V∗ + δV exp(λt)

n = n∗ + δn exp(λt)

• Linearize equations around fixed point, and obtain λs as eigenvalues of the Jacobian matrix

M =

(∂f∂V

∂f∂n

∂g∂V

∂g∂n

)

• If both λs have negative real parts, fixed point stable

• Bifurcation occurs when Re(λ) = 0 when a parameter (such as Iapp) is varied

Classification of fixed points

• Real eigenvalues:

– Two negative eigenvalues: stable node (sink node)

– One positive, one negative: saddle

– Two positive eigenvalues: unstable node (source node)

• Complex eigenvalues:

– Negative real part: stable spiral (spiral sink)

– Positive real part: unstable spiral (spiral source)

From resting to periodic firing in 2D models

Bifurcations leading to AP generation

A bifurcation occurs when at least one eigenvalue has zero real part. In a 2D model, two

scenarios:

• Two real eigenvalues, one is λ = 0⇒ Steady-state (saddle node) bifurcation;

• Two complex eigenvalues, λ = ±iω⇒ Hopf bifurcation;

Two types of saddle-node bifurcations from resting to periodic spiking

Two types of Hopf bifurcations from resting to periodic spiking

Relationship with Hodgkin (1948) classification

• Class 1 neural excitability (Type I): APs can be generated with arbitrarily low

frequency

– SNIC

• Class 2 neural excitability (Type 2): APs are generated with a frequency that cannot

be lower than fc > 0.

– Saddle node

– Hopf (both sub and supracritical)

• Class 3 neural excitability: only single APs, no repetitive firing.

Hopf: Morris-Lecar

CdV

dt= −gL(V − VL)− gCa(V − VCa)m∞(V )− gKn(V − VK) + Iapp

τn(V )dn

dt= −n+ n∞(V )

Phase plane of ML in the Hopf regime

Bifurcation diagram of ML in the Hopf regime

Bifurcation diagram of the HH model

Morris-Lecar in the SNIC regime

SNIC: Morris-Lecar

Class I vs Class II: Example of cortical neurons

Pyramidal cells Interneurons

Bibliography

• Ermentrout-Terman book, chapter 3

More on dynamical systems:

• Strogatz, Nonlinear dynamics and chaos

• Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and

Bifurcations of Vector Fields

• Kuznetsov, Elements of Applied Bifurcation Theory

• Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and

Bursting