Mathematical Neuroscience: from neurons to networks Part I Cortex - PowerPoint PPT Presentation

Mathematical Neuroscience: from neurons to networks Part I Cortex Steve Coombes School of Mathematical Sciences

Neurons: pyramidal cells

Hodgkin and Huxley (1950s) express (and subsequently fit) the dynamics of gating variables (representing membrane channels) using the mathematical language of nonlinear ODEs. Action potentials m/s

Active membrane models C d v g k m p k k h q k � d t = − k ( v − v k ) + I C k p k , q k ∈ Z v − membrane potential m k , h k − gating variables g k − conductances v k − reversal potentials 1 activating X ∞ ( v ) d X d t = X ∞ ( v ) − X τ x ( v ) inactivating 0 -60 -40 -20 0 20 v (mV)

Hodgkin and Huxley: ( v, m, n, h ) 40 v v 0 t Reduction: m → m ∞ ( v ) -40 ( n, h ) → ( n ∞ ( u ) , h ∞ ( u )) -80 0 40 80 120 160 I u ˙ u = 0 -40 C d v d t = f ( v, u ) + I d u -50 d t = g ( v, u ) ˙ v = 0 -60 v -80 -40 0 40 Method of equivalent potentials gives and in terms of HH model - Abbott and Kepler 1990 g f

Cortical model (slow firing) 0.3 v 0.2 0.1 0 -0.1 -1 0 1 -1 0 1 w v v 0.8 0.4 0 50 100 v 0 SNIC -0.4 0 250 500 t 0 0.2 0.4 I � Freq ∼ I − I c

Morris-Lecar model (slow firing) ( v, w ) Originally a model of the barnacle giant muscle fiber 0.2 v w 0.4 Freq Type I f 0.2 0.1 0 homoclinic 0 w -0.4 -0.2 0 0.2 0.07 0.075 I 0.08 v I 1 Freq ∼ − ln ( I − I c )

Phase Response Curve (PRC) A PRC tabulates the transient change in the cycle period of an oscillator induced by a perturbation as a function of the phase at which it is received. HH Cortical ML 40 0.6 2 0.2 800 1 Q v v 0.1 0 400 0.3 0 0.6 0 -40 0 0.2 0 -2 -0.1 -80 -400 -0.2 -0.3 -4 -0.2 0 T 0 T 0 T obtained numerically

Q = ∇ Z θ Isochrons as leaves of the stable manifold of a hyperbolic limit cycle Call the orbit where ˙ z = Z ( t ) z = F ( z ) Introduce a phase (isochronal coordinates) θ d Q D ( t ) = − DF T ( Z ( t )) d t = D ( t ) Q, ∇ Z ( 0 ) · F ( Z ( 0 )) = 1 and Q ( t ) = Q ( t + T ) T θ θ = 1 ˙ T

Weak Coupling z i ∈ R m θ i ∈ S 1 θ θ γ i ⊂ R m θ i = 1, . . . , N Uncoupled system has an ˙ z i = F ( z i ) + � G i ( z 1 , . . . , z N ) exponentially stable limit cycle γ i Direct product of hyperbolic limit cycles is a normally hyperbolic invariant manifold Drive θ i = 1 ˙ T + � � Q ( θ i ) , G i ( Γ ( θ )) � PRC

Coupled oscillator networks An example: N 1 � gap junction ( v j − v i ) N coupling j = 1 � T H ( θ ) = 1 Averaging gives � Q ( t ) , ( v ( t + θT ) − v ( t ) , 0 ) � d t T 0 Kopell and Ermentrout θ i = 1 T + � ˙ � H ( θ j − θ i ) N E = 1 j � v j Morris-Lecar N j E 0.1 0 -0.1 -0.2 1000 2000 3000 4000 t

Stability of phase-locked states 14 φ1 φ H H φ+β Morris-Lecar 10 φ+2β and gap jns φ+3β 6 β = 1/Ν φ 2 φ3 � H n e 2πinθ H ( θ ) = 2 Dim(Fix (Γ)) = 1 Dim(Fix (Γ)) = 3 n Bifurcations from maximally symmetric solutions to ones with smaller isotropy groups. eg. cluster states. -2 0 0.2 0.4 0.6 0.8 1 θ θ Synchrony λ = − � H � ( 0 ) Asynchrony λ n = − 2 π in � H − n 0.6 t v E 0.6 0.4 0.5 E 200 600 1000 1400 1800 v t t 0.2 w w 0.4 0.6

Heteroclinic cycles Winnerless networks Rabinovich et al. Dynamical principles in neuroscience, Rev. Mod. Phys., 78, 2006. Ashwin et al. SIADS, Dynamics on networks of cluster states for globally coupled phase oscillators, 6, 2007. Applications of weakly coupled oscillator theory to CPGs, robot control, ... Biorobotics lab at EPFL http://biorob.epfl.ch/

Strongly coupled synaptic networks PSP synaptic processing j α 2 t e − α t α 2 te −α t dendritic processing i time w ij η ∗ W ij η ∗ N � � η ( t − T m s i ( t ) = g s ( v s − v i ( t )) j ) W ij j = 1 m ∈ Z T m v i > 0, t > T m − 1 = inf { t | v i ( t ) > h, ˙ } i i

Integrate-and-fire neurons v ss d v d t = − v t ∈ ( T m , T m + 1 ) τ + A ( t ) , v θ subject to nonlinear reset v reset t φ φ φ 1 2 Periodic forcing gives 0 p : q mode-locked states Implicit map of firing times Arnol’d tongue structure dominated by non-smooth bifurcations

CML - discrete time IF � V i ( n + 1 ) = [ γ V i ( n ) + � W ij a j ( n )] Θ ( 1 − V i ( n )) j a i ( n ) = Θ ( V i ( n ) − 1 ) Mexican hat interaction

Network firing maps 1 α −1 U1 U2 −1 ε − 2 α 3 τ � e 2πim/3 U m ( t ) V x ( t ) + iV y ( t ) = ISI n = T n + 1 − T n Global heteroclinic bifurcation (N=3) m = 1 α = 20 α = 17 0.402 0.43 α = 14 α = 17 α = 20 ISI n + 1 Dn+1 Dn+1 ISI n + 1 0.42 0.4 Vy 0.41 0.4 0.398 0.398 0.4 0.402 0.4 0.41 0.42 0.43 Vx ISI n Dn Dn ISI n P C Bressloff and S Coombes 2000 Dynamics of strongly-coupled spiking neurons, Neural Computation, Vol 12, 91-129

Fits to data 30 A. 50Hz 30 B. 110Hz 30 C. 200Hz VCN stellate cell ISI n+1 (ms) 20 20 20 10 10 10 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 ISI n (ms) ISI n (ms) ISI n (ms) 30 D. 30 E. 30 F. Linear IF and threshold noise ISI n+1 (ms) 20 20 20 10 10 10 ISI n = T n + 1 − T n 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 ISI n (ms) ISI n (ms) ISI n (ms) J Laudanski et al. , Journal of Neurophysiology, 103, 2010 100 ms − 1 τ ( v − v L ) + κ Nonlinear IF τ e ( v − v κ ) / κ Layer V cortical pyramidal cell Badel et al. , Journal of Neurophysiology, 99, 2010

Regular spiking Chattering Fast spiking Intrinsically bursting v = | v | + I − a ˙ τ ˙ a = − a a reset threshold S Coombes and M Zachariou 2009, in Eugene Izhikevich 2008 Coherent Behavior in Neuronal Networks v (Ed. Rubin, Josic, Matias, Romo), Springer.

Absolute IF networks a ( T m ) → a ( T m ) + g a / τ a reset threshold Orbit and PRC in closed form (pwl system) Gap jn network: asynchronous state network averages � T N 1 v ( t + jT/N ) = 1 � ~ lim v ( t ) d t ≡ v 0 N T N →∞ time averages 0 j = 1 ˙ ˙ v = | v | − � v + I − a + � v 0 , a = − a/ τ a advanced-retarded ode - self-consistent periodic solution

Stability and bifurcations e-values as zeros of (using phase density formalism): PRC of splay � 1 E ( λ ) = e λ T R ( θ ) e λθ T d θ v ( λ ) + �λ T � LT of orbit 0 8 g a g a 6 synchronized bursting 4 2 asynchrony 0 0 0.2 0.4 0.6 0.8 1 g �

Books Physica D Special Issue Mathematical Neuroscience Vol 239, May 2010

Mathematical Neuroscience: from neurons to networks Part II Cortex Steve Coombes School of Mathematical Sciences

Brain and Cortex

Principal cells and interneurons Santiago Ramón y Cajal Eugene Izhikevich 1900 2008

Electroencephalogram (EEG) power spectrum α EEG records the activity of ~ 10 6 pyramidal neurons.

Population model WEE Firing rate activity f ( E ) PE E WEI PI WIE I h WII Firing rate activity f ( I ) E = − E + W EE g EE ( A + − E ) + W EI g EI ( A − − E ) + P E ˙ τ E I = − I + W II g II ( A − − I ) + W IE g IE ( A + − I ) + P I ˙ τ I

η ( t ) = α 2 t e − α t WEE PE Q η = δ E WEI � 2 � d 1 + 1 Q = PI d t α WIE I t 0 WII Qg jE = f ( E ) Qg jI = f ( I ) Steady state approximation E = E ( g EE , g EI ) I = I ( g II , g IE ) Qg = f g = η ∗ f f = f ( { g } )

Alphoid chaos (10 D) p o w e r frequency P I LLE ( S − 1 ) inhibition Shilnikov saddle-node route to chaos van Veen and Liley, PRL, 97 , 208101 (2006) excitation P E

Spatially extended models g = w ⊗ η ∗ f Simplest neural field model: Wilson-Cowan (‘72), Amari (‘77) f ( u ( x, t − | x − y | /v )) g = u ( x, t ) y x w ( | x − y | ) � ∞ � ∞ u ( x, t ) = d y w ( y ) d s η ( s ) f ( u ( x − y, t − s − | y | /v )) − ∞ 0

2D layers u ab = η ab ∗ ψ ab b � h a = u ab b p o w w ab e r a frequency � R 2 d r � w ab ( r , r � ) f b ◦ h b ( r � , t − | r − r � | /v ab ) ψ ab ( r , t ) =

Turing instability analysis E layer and I layer e i k · r e λ t Continuous spectrum det ( D ( k, λ ) − I ) = 0 [ D ( k, λ )] ab = � η ab ( λ ) G ab ( k, − i λ ) γ b γ = f � ( ss ) η = LT η � G = FLT w ( r ) δ ( t − r/v ) S Coombes et al., PRE, 76 , 05190 (2007)

Re λ ( k ) λ = ν + i ω 0.2 0 ω 2 4 λ ( k ) i ω c 6 8 0 6 − i ω c 4 6 2 4 0 2 0 a. 20 0 γ ν 15 Turing-Hopf 10 Hopf 5 0 0 2 4 6 8 10 12 v ( axonal speed )

Amplitude Equations (one D) Coupled mean-field Ginzburg–Landau equations describing a Turing–Hopf bifurcation with modulation group velocity of . O ( 1 ) ∂τ = A 1 ( a + b | A 1 | 2 + c � | A 2 | 2 � ) + d ∂ 2 A 1 ∂ A 1 ∂ξ 2 + ∂τ = A 2 ( a + b | A 2 | 2 + c � | A 1 | 2 � ) + d ∂ 2 A 2 ∂ A 2 ∂ξ 2 − Benjamin–Feir (BF) BF-Eckhaus instability t t x k k Coefficients in terms of integral transforms of and . w η

Applications to co-registered EEG/fMRI Bojak, I., Oostendorp, T. F., Reid, A. T., Kotter, R., 2009. Realistic mean field forward predictions for the integration of co-registered EEG/fMRI. BMC Neuroscience 10, L2.