BME 6938Neurodynamics

Instructor: Dr Sachin S Talathi

Recap Linear stability analysis Bifurcation-Definition and Terminology Normal Form Bifurcation Diagram Saddle Node Bifurcation Transcritical Bifurcation Pitchfork Bifurcation

One Dimensional Neuron models

The dynamics of fast variable is much faster than time scale for membrane dynamics

Ifast

Types of 1-dimensional neuron models Ifast: Can be inward or outward The gating variable can be activating

(turned on by depolarization) or deactivating (turned on by hyperpolarization)

We can classify four model types: Inward activating: Persistent sodium current (INa,p) Inward de-activating: Fast potassium current (IK) Outward activating: h-current (I) Outward de-activating: Potassium inactivation

current (IKir)

Summary of 4 models

Inward Outward

Activation INa,p IK

Inactivation Ih IKir

V1/2 K ER

INa,p 1.5 16 60

IK -44 8 -90

Ih -75 -5.5 40

IKir -80 -12 -80

Amplifying and Resonating Currents

AmplifyingI’(V)<0: Provide positive Feedback through interationwith membrane voltageEg: INa,p, Ikir

Resonating: Damp oscillationsby providing negative feedbackEg: IK, Ih

Positive feedback Negative feedback

Specific Example: INa,p model

Parameters: C=10; Im=0; gL=19; EL=-67; gNa=74; V1/2=1.5; K=16;ENa=60

Dynamical behavior of INa,p model

Bifurcation in INa-P model

Lets Verify this bifurcation diagram in XPPAUTO

Makings of action potential Model

Glimpse of Neurocomputational property Slow transition to spiking: Ghost of saddle

node

Pyramidal cell in ratVisual cortex

Im=30

Read sections 3.2-3.3.7 from Izhikevichs book

Specific Example: INa,p model

Parameters: C=10; Im=0; gL=19; EL=-67; gNa=74; V1/2=1.5; K=16;ENa=60

Dynamical behavior of INa,p model

Bifurcation in INa-P model

Lets Verify this bifurcation diagram in XPPAUTO

Makings of action potential Model

Glimpse of Neurocomputational property Slow transition to spiking: Ghost of saddle

node

Pyramidal cell in ratVisual cortex

Im=30

Read sections 3.2-3.3.7 from Izhikevichs book

XPPAUTO-Exercise Bifurcation analysis using AUTO. We will use the example provided in Ex2.ode

file. We will soon see that this example is 1d

version of the more famous Morris-Lecar Neuron Model.