Dynamical systems in neuroscience Pacific Northwest Computational Neuroscience Connection October 1-2, 2010
What do I mean by a dynamical system? Set of state variables Law that governs evolution of state variables in time Often takes the form of differential equations ex: dv dw = V ( a V )( V 1) w + I = bv cw
What do I mean by a dynamical system? Set of state variables Law that governs evolution of state variables in time Often takes the form of differential equations ex: dv = V ( a V )( V 1) w + I state variables dw = bv cw parameters
Dynamical systems arise as models for single neurons ex: Hodgkin-Huxley equations C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L ( ( V ) n ) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h ( ( V ) h) / τ ( h V )
Hodgkin and Huxley s circuit model of a neuronal membrane: 10
Hodgkin and Huxley s circuit model of a neuronal membrane: Voltage response to input current 10µA/cm 2 V(mV) 50-50 0 t (ms) 50 11
Hodgkin and Huxley s circuit model of a neuronal membrane: f(hz) 12
C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L ( ( V ) n ) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h ( ( V ) h) / τ ( h V ) dv = F( V) ( ) V = V,n, m,h Basic question: can I get info about ( ) solutions by analyzing F V?
Phase plane analysis 0.2 0.15 0.1 0.05 0-0.05-0.1-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 dv dw = V ( a V )( V 1) w + I = bv cw
Phase plane analysis: Nullclines 0.2 dw = 0 0.15 0.1 0.05 dv = 0 0-0.05-0.1-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 dv dw = V ( a V )( V 1) w + I = bv cw
Phase plane analysis: Equilibria and stability 0.2 0.15 0.1 0.05 0-0.05-0.1-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.4 1.2 1 0.8 0.6 0.4 0.2 0-0.2-0.4 0 20 40 60 80 100 dv dw = V ( a V )( V 1) w + I = bv cw a=.1, b=.01, c=.02, I=0
Phase plane analysis: Equilibria and stability 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0-0.05-0.1-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.2 1 0.8 0.6 0.4 0.2 0-0.2-0.4 0 50 100 150 200 250 300 dv dw = V ( a V )( V 1) w + I = bv cw a=.2, b=.01, c=.02, I=0
Phase plane analysis: Bistability 0.2 0.15 0.1 0.05 0-0.05-0.1-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.4 1.2 1 0.8 0.6 0.4 0.2 0-0.2-0.4 0 20 40 60 80 100 dv dw = V ( a V )( V 1) w + I = bv cw a=.1, b=.01, c=.07, I=0
Bifurcations: a qualitative change in 0.2 behavior 0.15 0.1 0.05 0-0.05 dv dw -0.1-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 = V ( a V )( V 1) w + I = bv cw I = 0
Bifurcations: a qualitative change in behavior 0.3 0.25 0.2 0.15 0.1 0.05 dv dw 0-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 = V ( a V )( V 1) w + I = bv cw I = 1
Bifurcation diagrams: an example V 1.5 1 V 0.5 0-0.5-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 I_0 I0
Two types of bifurcations: Type I vs. Type II (or Class 1 vs. Class 2) saddle node on invariant circle Class 1 excitability subcritical Hopf: Class 2 excitability firing rate firing rate Mean input I Mean input I
Some simplified models Phase model Integrate-and-fire Firing rate models
Reduction of 1. Brain recordings: voltage spikes via ionic currents neurons to phases: 2. Nonlinear oscillator eqn. for each neuron V q t V = voltage q = conductance e.g Rose and Hindmarsh, 1989. Hodgkin and Huxley, etc. 3. Coordinate change phase V θ fire V 0 2π θ Winfree 74, Guckenheimer 75, 25
Reduction of neurons to Nonlinear oscillator eqn. for each neuron phases: V = voltage q = conductance V q t 3. Coordinate change phase V θ fire V 0 2π θ Winfree 74, Guckenheimer 75, 25
Goal: simple phase description natural frequency original neural 26
Goal: simple phase description natural frequency original neural J(x,t) 27
Goal: simple phase description natural frequency original neural J(x,t) perturbation J(x,t) J(t) J(x,t) J(x,t) 28
Finding θ / V = z(θ), the phase response curve:, parameterized by θ NOTE: this technique often used in lab experiments! perturb with 5 mv stim. Thanks to Jeff Moehlis 30
Phase response curves for different neurons look very different! [Ermentrout and Kopell, Van-Vreeswick, Bressloff, Izhikevich, Moehlis, Holmes, S-B] Hodgkin-Huxley Leaky Integrate and Fire 33
With phase dynamics natural frequency phase response curve (phase sensitivity curve) π θ fire θ=0 31
We can study synchrony in network of two coupled neurons I syn (t) 32
For example, Hodgkin-Huxley... C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L ( ( V ) n) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h ( ( V ) h) / τ ( h V ) 37
The Hodgkin-Huxley model PRC z(θ) 38
The Hodgkin-Huxley model PRC z(θ) Moral: Fast excitatory coupling can synchronize HH neurons 39
Analyze via Poincare map between firing times of θ 1 Nancy Kopell, Bard Ermentrout, -- weak coupling theory See: synchronized state θ 12 =0 is stable fixed point for map 41
Let s try Hodgkin-Huxley + A-current C dv dn = n dm dh = I g K n 4 ( V E ) K g Na m 3 h( V E ) Na g ( L V E ) L g A a 3 b( V E ) A ( ( V ) n) / τ n V ( ) = ( m ( V ) m) / τ ( m V ) = h da = a db = b ( ( V ) h ) / τ h V ( ) ( ( V ) a ) / τ a V ( ) ( ( V ) b) / τ b V ( ) 42
The Hodgkin Huxley plus A current model PRC z(θ) 44
The Hodgkin Huxley plus A current model PRC z(θ) Moral: Excitatory coupling actually DEsynchronizes HH neurons with A currents Stable anti-synchronized state However, inhibition does synchronize! When inhibtion, not excitation, synchronizes Van Vreeswijk et al 1995 45
Beyond impulse coupling Brain has gap junctions, Kuramoto, Kopell, Ermentrout -average coupling functions: as well as slow chemical synapses. get a system depending on phase differences only 46
Integrate-and-fire models Hodgkin- Huxley V(t) Claim: conductances approx. constant in this range Above value V thresh, stereotyped spike takes over In range [V reset,v thresh ], following rescaling, V thresh V reset INTEGRATE AND FIRE MODEL OF LAPICQUE Gerstner, Abbott, others relate to HH V(t) 1 0 20
Many variations... dv = V + av 2 ; V < V thresh quadratic integrate and fire (QIF) dv = 1 τ m ( ( E V + Δ e V V T )/Δ T ) L T exponential integrate and fire (EIF) V(t) 1 0 20
Firing rate models David Tank 56
How does firing rate f depends on input current I? via different firing rate vs. current ( f-i ) curves Hodgkin-Huxley Integrate + Fire f(hz) f(hz) I I f(hz) Piecewise linear f(hz) Sigmoidal, gain g I I 57
Dynamics? Think of neural units described by firing rates y which approach equilibrium rates f(input) with time constant τ m. nonlinearity of input-output function f allows amazing array of neural network computations Hopfield, Grossberg, Cohen associative memory: recall and learning can encode complex input-output functions 58
Computational tools for dynamical systems XPP AUTO MATCONT
References Check website later! http://depts.washington.edu/compnsci/