Nonlinear Dynamics of Neural Firing BENG/BGGN 260 Neurodynamics University of California, San Diego Week 3 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 1 / 16
Reading Materials E.M. Izhikevich, Dynamical Systems in Neuroscience, MIT Press, 2007, Ch. 1-2, pp. 53-121. C. Koch, Biophysics of Computation, Oxford Univ. Press, 1999, Ch. 7, pp. 172-192. BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 2 / 16
One Dimensional Systems 1 m (V) C m dv m = I ext ḡ Na m (V m) (V m E Na ) }{{} g L (V m E L ) V 1/2 Vm I Na,p persistent, fast Na + (Izh. p.55) 1 m = 1 + e (Vm V 1/2)/k INa,p 0 V 1/2 Vm dv = F (V ) I ext F(V) 0 Vm BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 3 / 16
Phase Portrait, Equilibria dv = F (V ) df dv < 0 df dv > 0 df dv < 0 Small-signal analysis: V = V 0 + ṽ dṽ αṽ df with α = dv ṽ ṽ 0 e +αt V0 stable for α 0 (decaying perturbation) BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 4 / 16
Bistability and Hysteresis Figure 3.16: Membrane potential bistability in a cat TC neuron in the presence of ZD7288 (pharmacological blocker of I b ). (Modified from Fig. 6B of Hughes et. al. 1999). Figure 3.17: Bistability and hysteresis loop as I changes. Izhikevich, pg. 66 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 5 / 16
Saddle-Node Bifurcation Figure 3.30 Bifurcation diagram of the system in Fig. 3.26. Figure 3.25: Bifurcation in the I Na,p -model (3.5): The resting state and the threshold state coalesce and disappear when the parameter I increases. Izhikevich, pg. 72, pg. 77 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 6 / 16
Two-Dimensional Systems Morris-Lecar C m dv m = I ext ḡ K w (V E K ) ḡ Ca m (V ) (V E Ca ) g L (V E L ) n (n = w) Na Na I Na,p + I K ML dw n 4 = w (Vm) w τ w (V m) m 3 (1 n) simplified HH FitzHugh-Nagumo, etc. { dv = F (V, W ) dw = G (V, W ) Dynamic repertoire: Stationary Points Limit Cycles Stable Attractors No other limits in 2-D (since V and W are bounded) BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 7 / 16
Nullclines { dv = F (V, W ) dw = G (V, W ) Equilibrium Stable Stationary Point Unstable? BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 8 / 16
Stability Equilibria: { F (V 0, W 0 ) = 0 G (V 0, W 0 ) = 0 intersection of nullclines Linear Analysis: { V = V 0 + ṽ W = W 0 + w { dṽ = aṽ + b w d w = cṽ + d w with a = F V V0,W 0 b = F W c = G V V0,W d = G 0 Eigenanalysis: Jacobian ( ) ( ) ṽ a b U = A = w c d V0,W 0 W V0,W 0 eigenvectors du = A U, or U = c 1U 1 e λ1t + c 2 U 2 e λ2t with A U 1 = λ 1 U 1 A U 2 = λ 2 U 2 { Re λ 1 < 0 and Re λ 2 < 0 stable fixed point Re λ 1 > 0 or Re λ 2 > 0 unstable eigenvalues BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 9 / 16
Stability (Continued) Eigenanalysis: A U = λu det (A λi) = 0 a λ b c d λ = 0 λ 2 τλ + = 0 with λ 1,2 = τ ± τ 2 4 2 τ 2 τ 2 τ 2 { τ = a + d : Trace (A) = ad bc : det (A) > 4 : real, distinct eigenvalues = 4 : real, repeated eigenvalues < 4 : complex conjugate eigenvalues BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 10 / 16
Stability of 2-D Equilibria Figure 4.15: Classification of equilibria according to the trace (τ) and the determinant ( ) of the Jacobian matrix A. The shaded region corresponds to stable equilibria. Izhikevich, pg. 104 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 11 / 16
Morris-Lecar and I Na,p + I K Persistent sodium and potassium ( I Na,p + I K ) is equivalent to Morris-Lecar (n = w). V nullcline : F (V, n) = 1 ( I ext ḡ K n (V E K ) ḡ Na m (V ) (V E Na ) C ) g L (V E L ) = 0 n = Iext ḡ Nam (V ) (V E Na ) g L (V E L ) ḡ K (V E K ) n V-nullcline V n nullcline : G (V, n) = n (V ) n τ n (V ) n = n (V ) = 0 n n-nullcline V BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 12 / 16
I Na,p + I K (or ML) Nullclines Figure 4.4: Nullclines of the I Na,p + I K -model (4.1, 4.2) with low-threshold K + current in Fig. 4.1b. (The vector field is slightly distorted for the sake of clarity of illustration). Izhikevich, pg. 93 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 13 / 16
Graded Action Potentials Figure 4.7: Failure to generate all-or-none action potentials in the I Na,p + I K -model. Izhikevich, pg. 95 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 14 / 16
Variable Threshold Figure 4.8: Failure to have a fixed value of threshold voltage in the I Na,p + I K -model. Izhikevich, pg. 96 BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 15 / 16
Stable Limit Cycle Figure 4.10: Stable limit cycle in the I Na,p + I K -model (4.1, 4.2) with low-threshold K + current and I = 40. Izhikevich, pg. 97. BENG/BGGN 260 Neurodynamics (UCSD) Nonlinear Dynamics of Neural Firing Week 3 16 / 16