Simplified Models of Single Neuron Baktash Babadi Fall 2004, IPM, SCS, Tehran, Iran

References Koch (1999) Simplified models of single neurons. Rinzel & Ermentrout (1997) Analysis of neural excitability and oscillation. Gerstner & Kempter (2002) Spiking neural models Koch (1999) Phase Space Analysis of Neural Excitability

Why Simplified Models? Analysis of the dynamical behaviors of single neurons Reduce the computational load To be used in network models

From Compartmental models to Point Neurons Axon hillock

Point Neurons General Form :

Two Dimensional Neurons Enables phase plane analysis Most important variants Fitzhugh-Nagumo Model Fitzhugh-Nagumo Model Morris-Lecar Model Morris-Lecar Model Software: XPPAUT

Fitzhugh-Nagumo Model (1) A simplification of HH model m is much faster than the others: m is much faster than the others:

Fitzhugh-Nagumo Model (2) Eliminating the fast dynamics Blue: Full system Red: Reduced System

Fitzhugh-Nagumo Model (3) Eliminating h

Fitzhugh-Nagumo Model (4) Fitzhugh-Nagumo 2D model:

Fitzhugh-Nagumo Model (5) Nullclines: V nullcline n nullcline Orbit or Spiral?

Fitzhugh-Nagumo Model (6) Fitzhugh- Nagumo equations: V W Qualitatively capyures the properties of the exact model

Analysis of Fitzhugh-Nagumo System (1) Jacobian: Fixed point for I=0 : V=-1.20, W=0.625 The fixed point is: Stable Spiral

Response of the resting system (I=0) to a current pulse: Analysis of Fitzhugh-Nagumo System (2) W V

Response of the resting system (I=0) to a current pulse: Analysis of Fitzhugh-Nagumo System (3) W V Threshold is due to fast sodium gating (V nullcline) Hyperpolarization and its termination is due to sodium/potassium channel

Response to a steady current : Analysis of Fitzhugh-Nagumo System (4) Jacobian: Fixed point for I=1 : V=0.41, W=1.39 The fixed point is: Unstable spiral

Response of the resting system (I=1) to a steady current: Analysis of Fitzhugh-Nagumo System (4) W V Stable Limit Cycle

Response of the resting system (I=1) to a steady current: Analysis of Fitzhugh-Nagumo System (4) W V

Bifurcation By increasing the parameter I the stable fix point renders unstable I the stable fix point renders unstable A stable limit cycle appears A stable limit cycle appears If by changing a parameter qualitative behavior of a system changes, this phenomenon is called bifurcation and the parameter is called bifurcation parameter

Onset of oscillation with non-zero frequency In the resting (I=0) the fixed point is a stable spiral The imaginary part of the eigenvalue is not zero and the real part is negative In the bifurcation, the fixed point loses its stability  the real part of eigen value becomes positive and the imaginary part remains non-zero The frequency of oscillation is proportional to the magnitude of the imaginary part of the eigenvalue By increasing I, the oscillation onset starts with non-zero frequency Hopf Bifurcation

IF response of Fitzhugh-Nagumo model f I

Neuron Type I / Type II Gain functions of type I and Type II neurons III Type I : Axon Hillock of most neurons Type II: Axons of Most neurons, whole body of non-adaptive cortical interneurons, the spinal neurons Neural Coding

Question What happens if we feed the Fitzhugh- Nagumo neuron with a strong inhibitory pulse? What happens if we feed the Fitzhugh- Nagumo neuron with a strong inhibitory pulse? Post inhibitory rebound spike Post inhibitory rebound spike V W

Morris - Lecar Model : an example of type I neuron Originally proposed for barnacle muscles. Fast system is the dynamics of Ca ions Slow system is the dynamics of K ions

Morris-Lecar Model (1) V

Morris-Lecar Model (2) Stable Spiral

Morris-Lecar Model (3) Stable node Saddle node Unstable node Separatrix Current Pulse

Morris-Lecar Model (4) Stable node Saddle node Unstable node Saddle Node Bifurcation

IF response of Morris-Lecar model f I

Reminder HH Model is Type II Stevens-Conner Model is type I A hyper polarization activated current changes the dynamics A hyper polarization activated current changes the dynamics

Bursting Neurons Adding another slow process (Eugene Izhikevich 2000) Three dimensional phase plane Three dimensional phase plane

Integrate-and-Fire Neuron (1) Maybe the most popular neural model One of the oldest models (Lapicque 1907) Although very simple, captures almost all of the important properties of the cortical neuron Divides the dynamics of the neuron into two regimes Sub Threshold Sub Threshold Supra Threshold Supra Threshold

Integrate-and-Fire Neuron (2) Sub-Threshold: The HH equations show that in sub-threshold regime, sodium and potassium active channels are almost close The HH equations show that in sub-threshold regime, sodium and potassium active channels are almost close The corresponding terms can be neglected in the voltage equation The corresponding terms can be neglected in the voltage equation

Integrate-and-Fire Neuron (3) Sub Threshold: Linear ODE Linear ODE With on input ( I ext =0 )Stable fixed point at ( V=V l ) With on input ( I ext =0 )Stable fixed point at ( V=V l ) 2D modes also demonstrate this behavior: 2D modes also demonstrate this behavior: Stable node Saddle node Unstable node Separatrix

Supra threshold: The shape of the action potentials are more or less the same The shape of the action potentials are more or less the same At the synapse, the action potential events translate into transmitter release At the synapse, the action potential events translate into transmitter release  As far as neuronal communication is concerned, the exact shape of the action potentials is not important,  As far as neuronal communication is concerned, the exact shape of the action potentials is not important, rather its time of occurrence is important Integrate-and-Fire Neuron (4)

Supra Threshold: If the voltage hits the threshold at time t 0 : If the voltage hits the threshold at time t 0 : 1) a spike at time t 0 will be registered 2) The membrane potential will be reset to a reset value (V reset ) The system will remain there for a refractory period (t ref ) Integrate-and-Fire Neuron (5) t0t0 t V V th V reset

Integrate-and-Fire Neuron (6)

Integrate-and-Fire Neuron (7) Response to a Steady Current

Variants of Integrate-and-fire (IF) neuron Non-Leaky IF: Adaptive IF: Spike-Response model

Firing rate neurons Individual spikes are not modeled, rather the firing rate of the neurons are modeled. Individual spikes are not modeled, rather the firing rate of the neurons are modeled. Assumes that the information is coded by the firing rate of the neurons and individual spikes are not important

Firing rate neurons (1) The free membrane voltage equation (no threshold implemented): The firing rate is a function of the free membrane voltage: g is usually a monotonically increasing function. These models mostly differ in the choice of g.

Firing rate neurons (2) Wilson-Cowan model Usually two neurons (excitatory/inhibitory) Usually two neurons (excitatory/inhibitory) IE Input I Input E a b c d g(x) x

Firing Rate Neurons (3) Linear-Threshold model: Based on the observation of the gain function in cortical neurons: V f I f 100 Hz Physiological Range

Steady State model The input current and the firing rate of the neuron are constant in time: Used in Artificial Neural Networks (Perceptrons…)

Binary neuron Depending on the input, the neuron is either on (1) or off (0): Ising neuron (+1/-1): Used in Hopfield network Used in Hopfield network

Overview of the neural models: Detailed conductance based models (HH) Reduced conductance based models (Morris-Lecar) Two Dimensional Neurons (Fitzhugh-Nagumo) Integrate-and-Fire Models Firing rate Neurons (Wilson-Cowan) Steady-State models Binary Neurons (Ising) Biological Reality Artificial Numerical Simulation Analytical Solution