Simple models of neurons!! Lecture 4! 1!
Recap: Phase Plane Analysis! 2!
FitzHugh-Nagumo Model! Membrane potential K activation variable Notes from Zillmer, INFN 3!
FitzHugh Nagumo Model Demo! 4!
Phase Plane Analysis - Summary! Scholarpedia, FitzHugh Nagumo Model 5!
Recap HH model! 6!
Reducing HH models(2)! Original Reduced 7!
Reducing HH models(3)! 8! Dynamical Systems Neuroscience, Izhikevich
Reducing HH to a 2-D equation! g For all three variables! g First approximation: Replace m by its asymptoic value! Abbott and Kepler, Model Neurons: From HH to Hopfield 9!
Reducing HH to a 2-D equation! g 2-d model! g We want the time-dependence of U in f in reduced model to approximate the timedependence of F in the full model by changing h and n! 10!
Two-dimensional version of HH! 11!
0 Two-dimensional version of HH! 30 Limit cycle due to constant current injection (train of action potentials) 40 U 50 0 0 0 0 V - nullcline 60 0 20 0 70 20 40 100 50 0 50 V 20 12!
General HH Model Reduction Strategy! g Start with one equation for V and one for recovery variable (lets call it R)! g To match the models in phase space, the first equation has to be a cubic polynomial in V! [Wilson, Spikes, decisions, and actions, 1998]! 13!
General HH Model Reduction Strategy! g Start with one equation for V and one for recovery variable (lets call it R)!! g Null clines of this system of equations are:! [Wilson, Spikes, decisions, and actions, 1999]! 14!
General HH Model Reduction Strategy! Direct! reduction! Phase plane! reduction (eqn below)!! g Fitting to nullclines we get:! [Wilson, Spikes, decisions, and actions, 1999]! 15!
General HH Model Reduction Strategy! [Wilson, Spikes, decisions, and actions, 1999]! 16!
What is the code here?! g Intracellular recording from a locust projection neuron to 1s odor puffs! 60 mv 17!
One-Dimensional Reductions! g Perfect Integrate and Fire Model! I(t) C V $"(t # t i ) Whatʼs missing! In this model! compared to HH?! t ref dv (t) C dt V t = I(t) ( ) = V Thr " Fire+reset linear threshold 18!
One-dimensional Reductions! g Perfect Integrate and Fire Model! I(t) V $"(t # t i ) V Spike emission V Thr C reset t ref dv (t) C dt V t = I(t) ( ) = V Thr " Fire+reset linear threshold 19!
One dimensional reductions! g The successive times, t i, of spike occurrence:!! t i+1 " t i I(t)dt = CV th g Firing rate vs. input current of the perfect integrator:! I f = CV! Thr g If you force a refractory period T ref following a spike, such that V = 0mV for T ref period following a spike, then:! I f = CV Thr + t ref I 20!
One-dimensional Reductions! g Leaky Integrate and Fire Model! I(t) V $"(t # t i ) V Spike emission V Thr C R reset t ref dv (t) C dt V t + V (t) R = I(t) ( ) = V Thr " Fire+reset linear threshold 21!
Comparison of the two models! 10 9 8 Perfect Integrate & Fire Leaky Integrate & Fire Model Input 7 V(mV) 6 5 4 3 C = 1nF R = 10MOhm VSPK = 70mV VTHR = 15 mv I = 1 na What is the main! difference?! 2 1 0 0 50 100 150 200 250 300 350 400 450 500 Time (ms) 22!
C I(t) R One-dimensional Reductions! g Adapting Leaky Integrate and Fire Model! V $"(t # t i ) g adapt t ref C dv (t) dt " adapt dg adapt (t) dt + V (t) R + g adaptv (t) = I(t) linear = #g adapt (t) adaptation V ( t) = V Thr " Fire+reset threshold g adapt (t) = g adapt (t) + G inc 23!
Comparison of the three models! 300 250 Perfect Integrate & Fire Leaky Integrate & Fire Input Adapting Integrate & Fire 200 150 100 50 C = 1nF R = 10MOhm VSPK = 70mV VTHR = 15 mv τ G = 50 ms G inc = 0.2 ns I = 1nA 0 0 50 100 150 200 250 300 350 400 450 500 24!
Comparison of the three models! 250 200 Perfect I&F Leaky I&F Adapt I&F Firing rate (hertz) 150 100 50 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Input Amplitude (na) 25!
Low pass filter! 26!
One Dimensional Models! g A more principled approach (again based on 2-d phase plane dynamics!!!)! Input current moves! V nullcline! and! therefore alters stability of the! equilibrium point! y Slow Variable (W) 0.5 0 0.5 1 1.5 FitzHugh-Nagumo Model W-nullcline meets! left or right segments of! V-nullcline then the! equilibrium point is stable! W-nullcline meets! Center segments of! V-nullcline then the! equilibrium point is unstable! 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x Fast Variable (V) 27!
One Dimensional Models! g A more principled approach (again based on 2-d phase plane dynamics!!!)! Input current moves! V nullcline! and! therefore alters stability of the! equilibrium point! y Slow Variable (W) 0.5 0 0.5 1 1.5 Threshold behavior once the current alters! the stability of the fixed-point (causes spike)! FitzHugh-Nagumo Model W-nullcline meets! left or right segments of! V-nullcline then the! equilibrium point is stable! W-nullcline meets! Center segments of! V-nullcline then the! equilibrium point is unstable! 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x Fast Variable (V) 28!
One Dimensional Models! g A more principled approach (again based on 2-d phase plane dynamics!!!)! Sub-threshold dynamics captured by this highlighted region If I want to construct a integrate and fire model only this region is important!!! 29! Dynamical Systems Neuroscience, Izhikevich
Quadratic Integrate and Fire Model! g A more principled approach (again based on 2-d phase plane dynamics!!!)! Notice: in this highlighted region V-nullcline is a parbola U-nullcline is still a line Sub-threshold dynamics captured by this highlighted region 30! Dynamical Systems Neuroscience, Izhikevich
Izhikevich Model! g A simple model that captures the subthreshold behavior in a small neighborhood of the left knee (confined to the shaded square) and the initial segment of the upstroke of an action potential is given by:! where a, b, c, d are dimensionless parameters! 31!
Izhikevich Model! 32!
Izhikevich Model! 33!
I(t) Firing rate models! g The potential in a continuous firing rate unit has same dynamics as in a LIF neuron! V f= g(v) C R C dv (t) + V (t) dt R f = g(v ) = I(t) Sigmoidal function 34!