BME 6938 Neurodynamics

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BME 6938Neurodynamics

• Instructor Dr Sachin S Talathi

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Recap-2D linear dynamical system-Explicit solution
OR
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Recap- Eigen values for 2D system
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Classification Scheme for fixed points

• We saw how to get explicit solution to any
  2-dimensional linear ODE.
• However if we are interested in global properties
  of the trajectories explicit solutions are
  unnecessary
• The eigen values of A give us all the information
  about the stability properties of the fixed
  points of the system and we can devise
  classification scheme based on the eigenvalues
  for the system

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Classification Scheme for fixed points

• Case I saddle node
• Case II
• (a) either stable or
  unstable node
• (b) either stable or unstable spiral
• (c) border line between nodes
  and spirals
• Case III one of the eigenvalues
  is zero no isolated fixed point but a series of
  fixed points (centers)

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Summary of the classification scheme
Degenerate nodes
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Dynamics around fixed point based on previous
analysis
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Invariant Manifolds

• The eigenvectors of A corresponding to the cases
  with non-degenerate eigenvalues considered
  earlier represent invariant manifolds for the
  dynamical system.
• Eg. Lets say the phase space is 2 dimensional
  made up of dynamical variables x and y. If
  initial condition is on x-axis and the flow as
  the system evolves remains on x-axis then x-axis
  is the invariant manifold of the dynamical
  system.
• In other words, orbits that start on the manifold
  remain in it.

Refer to class notes for more details on
Invariant manifolds
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Examples of Invariant Manifold

• Invariant manifolds of saddle (1-Dimensional
  manifolds)
• Spirals (II-Dimensional manifolds)

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Linear Stability Analysis for Nonlinear
Two-Dimensional System
Evaluated at the fixed point (x, y)
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Few comments

• The Linearized system does represent the dynamics
  of the nonlinear system locally around the fixed
  point correctly (Stable manifold theorem or the
  Hartman-Grobman Theorem) when the real part of
  eigenvalues are non-zero.
• Fixed point in these case are referred to as the
  hyperbolic fixed points. The contribution from
  nonlinear higher order terms is negligible
  locally around hyperbolic fixed points.
• Nonhyperbolic fixed points are those for which
  the real part of eigenvalue is zero. They are
  more sensitive to higher order nonlinear terms
  eg Centers Bifurcation points etc..

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Stable and Unstable Manifolds of Saddle
Unstable manifold v1
Stable manifold v2
Stable manifold of saddle is also referred to as
the seperatrix since it separates the phase
plane into different regions of long term behavior
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Revisit a simple example
We saw the phase space for this system earlier
in our class. Lets Revisit and now Identify the
stable and unstable manifolds of the saddle node
(This time using XPPAUTO)
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Important of stable manifold of saddle in
Neurodynamics
Note Threshold is not a single voltage
value. It is a curve in Phase space defined by
the Stable manifold Of saddle
Ex3.ode with parameters for type 1
neuron-gtGenerate above figure
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Homoclinic and Heteroclinic Trajectories
A trajectory is homoclinic if it originates from
and terminates at the same equilibrium point
A trajectory is heteroclinic if it originates at
one equilibrium and terminates at a different
equilibrium point
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Heteroclinic and Homoclinic orbits in Neuron model
Homoclinic Orbit Ex3.ode (High Threshold fast
(tau0.152) K current)
Heteroclinic Orbit Ex3.ode (High threshold
K-current)
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Transition to Spiking
Saddle Node Bifurcation
Hopf Bifurcation
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Saddle Node Bifurcation in Phase Space
I3
I10
I4.51
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Hopf Bifurcation in Phase Space
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Minimal models for spiking

• Minimal (conductance based) model for neuronal
  dynamics is a model that satisfies the following
  two criteria
• It has a limit cycle attractor
• If one removes any current or gating variable,
  the model only has fixed point equilibrium
  attractors
• Any conductance based model is either a minimal
  model or can be reduced to a minimal model by
  appropriately removing the gating variables

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Constructing a Mimimal model for Neuronal Dynamics

• Mixture of one amplifying and one resonant gating
  variable with a leak current results in a minimal
  model
• Amplifying Gating Variable Provide positive
  feedback through the interaction with membrane
  voltage. Eg. Activation gating variable of sodium
  channel carrying inward current (m-gating in HH
  model)
• Resonant Gating Variable Provide negative
  feedback through the interaction with membrane
  voltage. Eg. Activation gating variable of the
  potassium channel carrying outward
  current(n-gating in HH model)

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Amplifying vs. Resonating Gate Variables

• Does the following make sense?
• To generate a spike we need
• fast positive feedback and
• slow negative feedback

What if we have slow positive feedback? Act as a
low pass filter No effect on fast frequencies
and amplifying slow fluctuations.
What if we have fast negative feedback? It will
damp input fluctuations. Stabilize the fixed
point of the system
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Minimal models

• Activation Inward (AI) Amplify
• Activation Outward (AO) Resonate
• In-Activation Inward (II) Resonate
• In-Activation Outward (IO) Amplify

Currents
Minimal Models
Total of 6 minimal models (and not 4)
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Commonly observed minimal models
INa,pIK model
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Dynamic repertoire for INa-pIK model
Low threshold K currents Hopf- bifurcation
High threshold K currents Saddle node
bifurcation
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INa,t Model
Mechanism for spiking
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Dynamic repertoire for INa-t model

• Note
• V- Nullcline looks like a
• cubic parabola
• 2. h-axis is flipped
• to preserve counter clock
• wise rotation
• 3. For right choice of parameters
• we can get both saddle node and
• Hopf Bifurcations

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Seminal paper by Rinzel and Ermentrout

• Read the paper by Rinzel and Ermentrout that
  explains most of the ideas we have studied so far
  for exploring the dynamics of neuronal models.
• Read sections 1 through 3 in the paper and
  reproduce the results using XPPAUTO (Homework)

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HH model Revisited

• HH Equation for membrane dynamics of giant squid
  axon

Parameter Values
gNa 120 mS/cm2
gK 36 mS/cm2
gL 0.3 mS/cm2
ENa 55 mV
EK -72 mV
EL -50 mV
Vm -60 mV
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Typical Bifurcation Structure in HH model
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Reducing HH model

• Original HH model is 4 dimensional.
• We have learned tools to study the dynamics
  exhibited by two dimensional neuron models
  (Remember the minimal models)
• Lets try to reduce the 4-D HH model to 2-D HH
  model without sacrificing the Bifurcation
  Structure around the transition to spiking state

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Reducing HH model

• Step 1. Note the m-gate dynamics is much faster
  than the h and n gate dynamics
• Replace

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Reducing HH model

• Step 2. Note
• Replace

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The Reduced HH model is
Parameter Values
gNa 120 mS/cm2
gK 36 mS/cm2
gL 0.3 mS/cm2
ENa 55 mV
EK -72 mV
EL -50 mV
Vm -60 mV
Does the reduced HH model represent a minimal
model?
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Lets look at the shapes of AP in full and reduced
HH model
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Hodgkin Classification of Excitability

• Class 1 (Type 1) Excitability
• Action potentials can be generated with
  arbitrarily low frequency. The Frequency
  increases by increasing the applied current
• Class 1 excitability is seen when the rest
  potential disappears through saddle node
  bifurcation (mostly true)
• Class II (Type II) Excitability
• Action potentials are generated in certain
  frequency bands that are relatively insensitive
  in the strength of applied current
• Class II excitability is seen when the rest
  potential disappears through Hopf bifurcation
  (again mostly true)

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Hodgkin Classification
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Bifurcations in 2D neuron models-Phase Space
Class I
Bistable
Class II