Integral Equation and PDE Neuronal Models:

Transcription

1 Integral Equation and PDE Neuronal Models: W. C. Troy Integral Equation and PDE Neuronal Models: p.1/51

2 Part 1. 1D Single Neuron Models. (i) Hodgkin-Huxley (1952), Fitzhugh-Nagumo (1961) Part 2. 1D and 2D Integral Eq. Models (i) 1D and 2D Bump Solutions and Waves Part 3. Synchrony (i) Bulk Oscillations. (ii) 1D Synchrony and 2D Synchrony Integral Equation and PDE Neuronal Models: p.2/51

3 Part 1: 1D models A. L. Hodgkin and A. F. Huxley, J. Phys. (1952) a 2R I = g k n 4 (V V k ) + G Na m 3 h(v V Na ) + g l (V V l ) + I 0 2 V x 2 V = C m t + I n t = α n (V )(1 n) β n (V ) m = α m (V )(1 n) β m (V ) t h t = α h (V )(1 n) β h (V ) V = transmembrane voltage; n, m, h are dimensionless quantitites associated with potsssium channel activation, sodium channel activation and sodium channel inactivation. Integral Equation and PDE Neuronal Models: p.3/51

4 Voltage Clamp Hodgkin and Huxley set I 0 = 0 and inserted a silver wire along the interior of the giant axon of the squid Loligo. Experiments on the voltage clamped axon allowed them to determine the functions and parameters in the equations: dv C m dt n t m t h t = g k n 4 (V V k ) + G Na m 3 h(v V m ) + g l (V V l ) = α n (V )(1 n) β n (V ) = α m (V )(1 n) β m (V ) = α h (V )(1 n) β h (V ) There is a unique rest state (V,m,n,h) (V 0,m 0,n 0,h 0 ) Integral Equation and PDE Neuronal Models: p.4/51

5 ( ).1(V 10) V α n (V ) = exp ( ), β V 10 n (V ) =.125 exp ( ).01(V 25) V α m (V ) = exp ( ), β V 25 m (V ) = 4 exp ( ) V 1 α h (V ) =.07 exp, β h (V ) = 20 exp ( ) V Integral Equation and PDE Neuronal Models: p.5/51

6 Traveling Waves Let (V,m,n,h) = (V (z),m(z),n(z),h(z)), z = x ct, and solve a 2R I = g k n 4 (V V k ) + G Na m 3 h(v V Na ) + g l (V V l ) + I 0 d 2 V dz 2 dv = cc m dz + I dn dt = α n (V )(1 n) β n (V ) dm = α m (V )(1 n) β m (V ) dt dh dt = α h (V )(1 n) β h (V ) lim (V (z),m(z),n(z),h(z)) = (V 0,m 0,n 0,h 0 ) z ± Integral Equation and PDE Neuronal Models: p.6/51

7 Proofs (1) S. P. Hastings, "On travelling wave solutions of the Hodgkin-Huxley equations," Archive for Rational Mechanics, 60 (1976), (2) G. A. Carpenter, "A geometric approach to singular perturbation problems with applications to nerve impulse equations." Journal of Differential Equations, 23 (3) (1977), (3) G. A. Carpenter, "Periodic solutions of nerve impulse equations." Journal of Mathematical Analysis and Applications, 58(1) (1977), (4) G. A. Carpenter, "Bursting phenomena in excitable membranes." SIAM Journal on Applied Mathematics, 36 (1979), Integral Equation and PDE Neuronal Models: p.7/51

8 Open Problems: (a) prove that there is a second traveling wave solution satisfying a 2R I = g k n 4 (V V k ) + G Na m 3 h(v V Na ) + g l (V V l ) + I 0 d 2 V dz 2 dv = cc m dz + I dn dt = α n (V )(1 n) β n (V ) dm = α m (V )(1 n) β m (V ) dt dh dt = α h (V )(1 n) β h (V ) lim (V (z),m(z),n(z),h(z)) = (V 0,m 0,n 0,h 0 ) z ± (b) Prove (based on numerical studies) that the second solution is unstable. Integral Equation and PDE Neuronal Models: p.8/51

9 Fitzhugh-Nagumo Model The Fitzhugh-Nagumo System (1961) was derived as a simplification of the Hodgkin-Huxley model. u t w t = 2 u x 2 + u(u a)(1 u) + w + I 0 = b(u γw) u = transmembrane voltage; w = recovery variable. (1) R. FitzHugh, "Impulses and physiological states in theoretical models of nerve membrane." Biophysical J. 1 (1961), (2) J. Nagumo, S. Arimoto and S. Yoshizawa, "An active pulse transmission line simulating nerve axon." Proc. IRE. 50 (1962), Integral Equation and PDE Neuronal Models: p.9/51

10 Traveling Waves Let I 0 = 0 and (u,v) = (u(z),v(z)), z = x ct, and solve c du dz w t = d2 u + u(u a)(1 u) + w dz2 = b(u γw) lim (u(z),v(z)) = (0, 0) z ± (1) S. P. Hastings, "On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations," The Quart. J. Of Math., (1976), Oxford Univ. Press (2) S. P. Hastings, "Single and Multiple Pulse Waves for the FitzHugh-Nagumo Equation,", SIAM J. Appl. Math 42 (1982), Integral Equation and PDE Neuronal Models: p.10/51

11 Stability of Wave Fronts General problem: u t = u xx + f(u), < x < f(0) = f(1) = 0, f (0) < 0,f (1) < 0, Wave fronts: u = U(z), z = x ct solves 1 0 f(u)du > 0 cu (z) = U (z), U( ) = 0, U( ) = 1 Fife and McLeod prove that that a wide range of initial conditions evolve into a wave front as t. (1) P. Fife and J. B. McLeod "The approach of solutions of nonlinear diffusion equations to travelling front solutions," Arch. Rat. Mech. and Anal. 65 (1977), Integral Equation and PDE Neuronal Models: p.11/51

12 Linearized Stability of Full Wave Let (U(z),V (z)) denote a traveling wave solution of the FHN Eqs. Linearize around (U(z),V (z)) : c dp dz dq dz = d2 p dz 2 + f (U) + q, f(u) = u(1 u)(u a) = b(p γq), 0 < b << 1 (1) C. Jones, "Stability of the traveling wave solution of the Fitzhugh-Nagumo system," Trans of AMS.286 (1984), Integral Equation and PDE Neuronal Models: p.12/51

13 Current Clamp HH Eqs Apply current I 0 to the voltage clamped HH system: C m V (t) = g k n 4 (V V k ) + G Na m 3 h(v V m ) n t m t h t + g l (V V l ) + I 0 = α n (V )(1 n) β n (V ) = α m (V )(1 n) β m (V ) = α h (V )(1 n) β h (V ) Rest state: (V,m,n,h) (V I0,m I0,n I0,h I0 ) for each I 0. Okan Gurel (1973) predicted that a Hopf bifurcation of periodic solutions occurs at a critial value I crit Integral Equation and PDE Neuronal Models: p.13/51

14 Current Clamp FHN Model Apply current I 0 to the voltage clamped FHN system: du dt w t = u(u a)(1 u) + w + I 0 = b(u γw) Rest state: (u,v) (u I0,v I0 ) for each I 0. Goal: prove that periodic solutions in an interval of I 0 values. Integral Equation and PDE Neuronal Models: p.14/51

15 Theory Theorem (WCT 1974) For the HH Eqs., and also for the FHN system, there is a critical value I crit such that a family of periodic solutions bifurcates from the rest state as I 0 passes through I crit. Proof for FHN: Apply the Hopf Bifurcation Theorem. (I) Linearize the system around the rest state (u I0,v I0 ) and let λ = α ± iβ denote the associated eigenvalues. (II) Show that α = 0 and β 0 at a critical value I 0 > 0. Integral Equation and PDE Neuronal Models: p.15/51

16 Experimental Verification The prediction of Okan Gurel was verified in Guttman et al, "Control of repetitive firing in squid axon membrane as a model for a neuron oscillator.: J Physiol (1980) 305: Integral Equation and PDE Neuronal Models: p.16/51

17 Part 2. Large Scale Models Integral equation systems were derived to model the behavior of populations of connected excitatory and inhibitory neurons. (A) Wilson-Cowan (1973) (B) Amari (1977) (C) Pinto-Ermentrout (2001) Integral Equation and PDE Neuronal Models: p.17/51

18 Wilson-Cowan Model 1973 E t I t = E + ω EE (x x,y y )f(e θ 1 )dx dy R 2 α ω IE (x x,y y )f(i θ 2 )dx y + Ψ 1 (x,y,t) R = I + ω EI (x x,y y )f(e θ 1 )dx R γ ω II (x x,y y )f(i θ 2 )dx + Ψ 2 (x,y,t) R E and I denote average activity of excitatory and inhibitory neuronal populations, ω ij = connection functions, f = firing rate, θ i = threshold, ψ i (x,y,t) = external input. H. R. Wilson and J. D. Cowan, Kybernetik 13 (1973), Integral Equation and PDE Neuronal Models: p.18/51

19 Amari Model 1977 τ u t = u + R ω(x x )f(u(x ) θ)dx + s(x,t) u = average activity of an excitatory neuronal population ω = the connection function, s(x,t) = external input { 0 if u theta, f(u) = 1 if u > θ Solutions: stationary bumps - regions of excitation, waves S. Amari, Biological Cybernetics 27 (1977), Integral Equation and PDE Neuronal Models: p.19/51

20 McLeod-Ermentrout 1993 Single population of excitable neurons: u t = u + k(x x )S ( u(x,t) ) dx k L 1 (, ),k 0 is even, k(η)dη = 1 S C 1 [0, 1],S 0,S(0) = 0,S(1) = 1,S (0) < 1,S (1) < 1 Theorem There is a unique monotonic traveling wave solution u(z),z = x ct such that u( ) = 0 and u( ) = 1 Integral Equation and PDE Neuronal Models: p.20/51

21 PDE (Laing and Troy 2003) u t + u = R 2 w ( (x s) 2 + (y q) 2 ) f(u(s,q,t) θ)dsdq Apply the Fourier transform F(g) (2π) 1 R2 exp( i(αx + βy))g(x,y)dxdy F(u + u t ) = F(w) F (f(u θ)) If w = w(r) then F(w) = F ( α 2 + β 2 ). To obtain the PDE approximate F(w) by a rational function of α 2 + β 2. Integral Equation and PDE Neuronal Models: p.21/51

22 F(u + u t ) = F(w) F(f(u θ)) F(w) = A B + (α 2 + β 2 M) 2 ( (α 2 + β 2 ) 2 2M(α 2 + β 2 ) + B + M 2) F(u+ut ) = A F(f(u θ)) Identities: (α 2 + β 2 ) 2 F(g) = F( 4 g) and (α 2 + β 2 ) F(g) = F( 2 g) Resulting PDE: ( 4 + 2M 2 + B + M 2) (u t + u) = Af(u θ) Integral Equation and PDE Neuronal Models: p.22/51

23 Example Let w = 2.1e r. Find M,A,B that minimize F(w) A B + (α 2 + β 2 M) 2 M = 2.5, A = 7, B =.52 w Circles: w = 2.1e r Solid curve: approximation r Integral Equation and PDE Neuronal Models: p.23/51

24 The Firing Rate f(u θ) = 2exp ( ) ρ (u θ) 2 H(u θ) H is the Heaviside function, ρ =.1, θ = 1.5 f(u) u Integral Equation and PDE Neuronal Models: p.24/51

25 Pinto-Ermentrout System u t v t = u + = ǫ(βu v) w(x,y,q,s)f(u θ)dq ds v + I u(x,y, 0) = f(x,y, 0), v(x,y, 0) = 0 v is a recovery variable, I is external input Pinto and Ermentrout: I and II (2001); 1D waves, I 0. Folias and Bressloff (2004); 2D breathers, I = I(x,y) 0. Y. Guo and C.C. Chow, (2005); 1D standing pulses Integral Equation and PDE Neuronal Models: p.25/51

26 Complex Eigenvalues The linearization of u t v t = u + = ǫ(βu v). around the steady state (u,v) = (0, 0) is w(x,y,q,s)f(u θ)dq ds v du dt dv dt = u v = ǫ(βu v). Integral Equation and PDE Neuronal Models: p.26/51

27 The linearized system du dt dv dt has complex eigenvalues = u v = ǫ(βu v) µ ± = (ǫ + 1) ± i 4βǫ (ǫ 1) 2 2 if and only if β > β = (ǫ 1)2 4ǫ. Integral Equation and PDE Neuronal Models: p.27/51

28 ǫ = θ =.1, β = II. Waves: β = < β < β 8.5 No Bulk oscillations. Multi ring waves. Periodic waves. Rotating waves. Integral Equation and PDE Neuronal Models: p.28/51

29 Equivalent PDE s w(x,y,s,q) = w ( (x q) 2 + (y s) 2 ) g(q, s) The PDE method: transform into u t v t = u + = ǫ(βu v) w(x,y,q,s)f(u θ)dq ds v + I ( 4 + 2M 2 + B + M 2) (u + u t + v I) = Ag(x,y)f(u θ) v t = ǫ(βu v). Integral Equation and PDE Neuronal Models: p.29/51

30 Waves ǫ = θ =.1, β = 3 double ring periodic Integral Equation and PDE Neuronal Models: p.30/51

31 Rotating Waves Rabbit cortex: Petsche et al (1974) Chic retina: Gorelova & Bures (1983) Turtle: Prechtel et al (1997) Mouse Hippocampus: Harris-White et al (1998) EEG Patterns: jirsa/imaging.html Models RD Eqs., Targets and Spirals, Koppel & Howard (1973) BZ Model: Winfree (1974) Integrate and Fire Models: Chu, Milton & Cowan (1994) Theta Neuron Model: Osan & Ermentrout (2001) Integral Equation and PDE Neuronal Models: p.31/51

32 Rotating Wave 1 Time course of solution at (x,y)=( 3.8,1.8) u( 3.8,1.8,t) t β = 3 Spiral Drift Integral Equation and PDE Neuronal Models: p.32/51

33 Inhomogeneous Coupling ( 4 + 2M 2 + B + M 2) (u + u t + v) = Ag(r,θ)f(u θ) v t = ǫ(βu v). ǫ = θ =.1, β = 3, g =.5 (1 + exp(.02rsin(.5φ))) u(r, 0) = Kexp(.5r 2 ), v(r, 0) 0 Spiral Initiation: Method 2 Integral Equation and PDE Neuronal Models: p.33/51

34 Rotating Waves In Rat Brain Huang, Troy, Ma, Schiff, Yang, Laing, Wu (2004) Hunag, Xu, Takagaki, Gao, Wu (2010) Integral Equation and PDE Neuronal Models: p.34/51

35 In-Vivo Rotating Waves In Cat Brain Viventi et al (2011) - rotating waves were initiated when picrotoxin was applied next to 15mm by 15mm sensors. Integral Equation and PDE Neuronal Models: p.35/51

36 Part III. Synchrony: β β Coexistence of (i) bulk synchronous oscillations, (ii) stable one bump waves, (iii) stable rest state. Synchrony in one dimension. Synchrony in two dimensions. Integral Equation and PDE Neuronal Models: p.36/51

37 Bulk oscillations Time dependent solutions (u(t), v(t)) of u t = u +.5 v t = ǫ(βu v). exp( x x )H(u(x,t) θ)dx v, satisfy du dt dv dt = u + H(u θ) v = ǫ(βu v) Integral Equation and PDE Neuronal Models: p.37/51

38 Theorem 0.1 Let 0 < ǫ < 1. If θ > 0 is small there is a β > β such that if β β then du dt dv dt = u + H(u θ) v = ǫ(βu v) has a periodic solution which coexists with the stable rest state (u,v) = (0, 0). If ǫ = θ =.1 then β = Integral Equation and PDE Neuronal Models: p.38/51

39 ǫ = θ =.1 v v =0.6 u =0 θ u =0.3 β=12.6 u.4 θ 0 u v 15 t u =0 v.9 v =0 u =0 θ.5 β=β =12.61 u.4 θ 0 u v 9 t v v =0.6 u =0 θ u =0.3 β=15 u.4 θ 0 u v 9 t Integral Equation and PDE Neuronal Models: p.39/51

40 Synchrony in one dimension 1 u u(x,0)=.15e x2 1 u u(x,0)=.6e x 2 θ 0 θ 0 β=12.61 β= x 1 x Integral Equation and PDE Neuronal Models: p.40/51

41 1 u u(x,0)=.6e x2 t=0 1 u t=2.3 θ 0 θ 0 β= x x 1 u t=t 1 =6.4 1 u t=t 7 =49.3 θ θ x 1 50 L 0 L 50 x 1 u t=t 19 =135 1 u t= θ θ 1 L L x 1 L x L Integral Equation and PDE Neuronal Models: p.41/51

42 Spread of synchrony How fast does the region of synchrony [ L,L] expand outwards from the point of stimulus? 1 u t=t 7 =49.3 u Inset t=49.3 θ θ 0 0 x 1 50 L 0 L 50 x.1 L 2L= L=22.9 Integral Equation and PDE Neuronal Models: p.42/51

43 Behavior at the center u(0,t) x=0.4 θ t u(0,t).4 t 1 =6.4 u(0,t) θ Inset θ t t= t.4.1 Integral Equation and PDE Neuronal Models: p.43/51

44 Rate of expansion 0.5 u 0 t θ 0.5 T 0 3 t 6 ( ) t Rate of expansion = c T where c is traveling wave speed. c = 22.4 mm/s Wilson et al, Nature (2001) Integral Equation and PDE Neuronal Models: p.44/51

45 Two Dimensions u t = u v + R 2 2.1e (x x ) 2 +(y y ) 2 H(u(x,y,t) θ)dx dy, v t = ǫ(βu v), u(x,y, 0) = u 0 (x,y), v(x, 0) = 0 (x,y) R 2, For this coupling β = 8.5 Integral Equation and PDE Neuronal Models: p.45/51

46 β(x,y) = { 8.5 if (x,y) Ω1 = {(x,y) (x + 20) 2 + y 2 < 25 2 }, 7.0 otherwise. u(x,y, 0) = e (x+20) 2 +y 2 and v(x,y, 0) = 0, 0 x 2 +y Integral Equation and PDE Neuronal Models: p.46/51

47 Seizure Recordings Leo Towle, Jean-Paul Spire, John Milton: Electrocortigraphic readings of seizures from a grid of electrodes implanted on the cortical surface Integral Equation and PDE Neuronal Models: p.47/51

48 Seizure Propagation mm s Predicted Migration Rate = ( ) t T c 5.33 mm s Experimental Migration Rate 4 mm s Integral Equation and PDE Neuronal Models: p.48/51

49 Synchronization in one region can trigger synchronization in another. β(x,y) = 8.5 if (x,y) Ω 1 = {(x,y) (x + 20) 2 + y 2 < 12 2 }, 9.5 if (x,y) Ω 2 = {(x,y) (x 20) 2 + y 2 < 12 2 }, 7.0 otherwise. u(x,y, 0) = e (x+20) 2 +y 2 and v(x,y, 0) = 0, 0 x 2 +y Integral Equation and PDE Neuronal Models: p.49/51

50 Synchronization in one region inhibits synchronization in another if they are too close. β(x,y) = 8.5 if (x,y) Ω 1 = {(x,y) (x + 18) 2 + y 2 < 18 2 }, 9.5 if (x,y) Ω 2 = {(x,y) (x 18) 2 + y 2 < 18 2 }, 7.0 otherwise. u(x,y, 0) = e (x+18) 2 +y 2 and v(x,y, 0) = 0, 0 x 2 +y Integral Equation and PDE Neuronal Models: p.50/51

51 Conclusions The mean field modeling approach gives mathematical predictions which are (I) qualitatively meaningful, and (II) might be practically useful. Integral Equation and PDE Neuronal Models: p.51/51