mechanisms of action potential generation: 2d...
Post on 06-Mar-2021
6 Views
Preview:
TRANSCRIPT
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
top related