Single neuron models L. Pezard Aix-Marseille University
Biophysics Biological neuron Biophysics Ionic currents Passive properties Active properties Typology of models Compartmental models Differential models Hybrid models Discrete state automata Models of synapses Models of synapses
Biological neuron
The membrane Models of the electrophysiology of excitability
Electro-diffusion laws Fick s law: Ohm s law: Einstein s relation: J d = D c x J e = µzc V x D = µk BT q
Nernst-Planck equation Ionic form: Molar form: Current density: J = (µzc V x + µk BT q J = J/N = (uzc V x + u RT F c x ) c x ) I = J zf = (uz 2 Fc V x + uzrt c x )
Nernst s equation At equilibrium (I=0): V 2 V 1 = RT zf ln(c 2 c 1 ) Conventions of electrophysiology: 1. V m = V i V e = V 2. Positive currents: from inside to outside Equilibrium potential for an ion (k): E k = V (I k = 0) thus E k = RT zf ln(c e c i )
Ionic distributions Cell of Mammal (T = 37 o C): Gradients maintenance: Ions c i c e E k K + 140 5 90 mv Na + from 5 to 15 145 60 to 90 mv Cl 4 110 90 mv Active processes: pump (ATP-ases), exchanger, co-transporters Donnan s equilibrium Membrane potential is the result of a stationary regime of exchanges based on transmembrane ionic currents
Ionic currents Ionic channels: Models: Characteristic: Non-gated channels Gated channels Electrodiffusion Barriers models State models (kinetics and stochastics) I-V curve
Goldman-Hodgkin-Katz model Hypothesis: Current I(V): 1. Electrodiffusion in the membrane 2. No interaction between ions 3. Constant electric field (dv /dx = V /l) I k = P k zfξ c i c e e ξ 1 e ξ et ξ = zfv RT
I-V curves for GHK model 0 10 20 I(a.u.) 30 40 50 60 c o /c i = 1 c o /c i = 5 c o /c i = 10 c o /c i = 20 70 80 100 80 60 40 20 0 20 40 60 80 100 V (mv )
Ohmic (linear) model Hypothesis: Equivalent circuit: 1. Equivalent circuit 2. Currents follow Ohm s law V e ρ k E k Ik V i V m Current I (V ): I k = γ k (V E k ) with γ k = 1/ρ k
Resting potential V o G K G Na G Cl Linear model: E K I K E Na I Na I m V i E Cl I Cl V r = G NaE Na + G K E K + G Cl E Cl G Na + G K + G Cl Constant field (GHK): V r = RT ( ) F ln PK [K] i + P Na [Na] i + P Cl [Cl] e P Na [Na] e + P K [K] e + P Cl [Cl] i I m = I Na + I K + I Cl = 0
Passive and active currents Definitions: Formalism: 1. Passive currents have a constant conductance 2. Active currents have a variable conductance (voltage, stimulus or chemical sensitive). They are the basis of cell excitability. 1. Passive: 2. Active: I k = g k (V E k ) I k = ḡ k p( )(V E k )
Cable equation (Rall, 1957-1969) V o (x) I o (x) r o dx V o (x + dx) R m R m C m dx V r C m dx V r I ion dx I ion dx I t dx I t dx V i (x) I i (x) r i dx V i (x + dx) τ m V t +R mi ion = λ 2 m 2 V x 2 with: τ m = R m C m and λ m = Rm d 4R c
Temporal properties R V o E rest I R C I C τ dv dt = V r + RI V with τ = R m C m V (t) = V r +RI (1 e t/τ ) and R = R m /S Small I m V i Vm (mv) Big Vm (mv) Fast Slow Time Time
Spatial properties I o (x) r o r m Vr r m Vr V r I r dx x I i (x) r i x + dx I r dx Vm (mv) Small Large λ 2 d 2 V dx 2 = V (x) V r Space V m (x) = V r + V e x/λ
Sodium gated channel
Formalism for gated channels I k = ḡ k p( )(V E k ) = ḡ k m a h b (V E k ) Probability of opening for activation (m) and inactivation (h) gates. Number of activation (a) and inactivation (b) gates Independence of the gates: p( ) = m m m h h h }{{}}{{} a times b times Open / activated channel: Closed channel: m = 1 (partially activated: 0 < m < 1) and h = 1. Not activated: m = 0 and h = 1 Inactivated: m = 1 and h = 0
Action potential Hodgkin & Huxley (1952): Experimental analysis and theoretical model that explains the genesis of action potential in the squid giant axon. Hypothesis: 1. No spatial dependency 2. Ohmic currents 3. Sodium, potassium and leak 4. First order kinetics for activation and inactivation gates
First order kinetics Closed α β Open df dt = α(1 f ) βf f (t) = f (1 e t/τ ) with f = α α + β and τ = 1 α + β f f(t) τ Time
Circuit for Hodgkin and Huxley model V o G L E L I L G K E K I K G Na E Na I Na C I C I = I Na + I K + I L + I C = I Na + I K + G L (V E L ) + C dv dt I m V i Active currents: Transient sodium current: I Na = ḡ Na m 3 h(v E Na ) Persistent potassium current: I K = ḡ K n 4 (V E K )
Historical Hodgkin-Huxley model C V = I ḡ Na m 3 h(v E Na ) ḡ K n 4 (V E K ) g L (V E L ) and ṁ = α m (V )(1 m) β m (V )m ḣ = α h (V )(1 h) β h (V )h ṅ = α n (V )(1 n) β n (V )n α n (V ) = 0.01 (10 V )/(exp((10 V )/10) 1) β n (V ) = 0.125 exp( V /80) α m (V ) = 0.1 (25 V )/(exp((25 V )/10) 1) β m (V ) = 4 exp( V /18) α h (V ) = 0.07 exp( V /20) β h (V ) = 1 / (exp((30 V )/10) + 1)
Potential dependence of kinetics parameters Time constant Asymptotic value 8 1 τ (ms) 6 4 2 τ m τ h τ n w 0, 8 0, 6 0, 4 0, 2 m h n 0 100 50 0 50 V (mv) 0 100 50 0 50 V (mv) τ x (V ) = 1 α x (V ) + β x (V ) x (V ) = α x (V ) α x (V ) + β x (V )
Hodgkin-Huxley model 50 V 0 50 100 0 10 20 30 40 50 60 70 80 90 100 Time 1 30 0, 8 P 0, 6 0, 4 h n G 20 10 g Na g K 0, 2 0 0 3 6 9 12 Time m 0 0 3 6 9 12 Time
Standard Hodgkin-Huxley model C V = I ḡ Na m 3 h(v E Na ) ḡ K n 4 (V E K ) g L (V E L ) ṁ = (m (V ) m) / τ m (V ) ḣ = (h (V ) h) / τ h (V ) ṅ = (n (V ) n) / τ n (V )
Standard kinetic parameters x (V ) 1 2k τ x (V ) C b + C a 0.5 0 x (V ) V 1/2 1 V 1 + e V 1/2 V k C b σ V max (Vmax V )2 τ x (V ) C b + C a e σ 2 V
Zoology of ion channels
Neuronal models Criteria: Categories: 1. Biological plausibility 2. Computational load 1. Compartmental models 2. Differential models (simplified HH) 3. Hybrid models 4. Discrete state automata
Compartmental models Principle:
Compartmental models Simulation environments: NEURON, GENESIS
Zoology of channels Channels types: Potassium channels: INa(fast), INa(slow), ICa(L), ICa(T), ICa(N), ICa(P), IK(DR), IK(A), IK(D), IK(M) IQ, Ih, If, IK(IR), ICl(V) IK(C), IK(AHP), ICl(Ca) IK(L), ICl, IK(ATP), IK(Na)
Reduction of Hodgkin-Huxley model: step 1 Separation of time scales: m = m (V ) τ (ms) 8 6 4 τ m τ h τ n P. 0.63 m n h 2 0 100 50 0 50 V (mv) Thus: τ m τ n τ m Time C V =I ḡ Na m (V ) 3 h(v E Na ) ḡ K n 4 (V E K ) g L (V E L ) ḣ =(h (V ) h)/τ h (V ) and ṅ = (n (V ) n)/τ n (V )
Comparison of HH models: 4D vs. 3D 50 1 0, 8 V 0 50 0, 6 P 0, 4 0, 2 100 0 10 20 30 40 50 60 70 80 90 100 Time 1 0 0 10 20 30 40 50 60 70 80 90 100 Time 1 0, 8 0, 8 P 0, 6 0, 4 P 0, 6 0, 4 0, 2 0, 2 0 0 10 20 30 40 50 60 70 80 90 100 Time 0 0 10 20 30 40 50 60 70 80 90 100 Time
Reduction of Hodgkin-Huxley model: step 2 Comparison of h and n : h + n 0.8 n + h 0, 85 0, 825 0, 8 Thus: 20 30 40 50 60 70 80 90 100 Time h 0, 75 0, 5 0, 25 0 0 0, 8 h = 0.89 1.1n n C V =I ḡ Na m (V ) 3 (0.8 n)(v E Na ) ḡ K n 4 (V E K ) g L (V E L ) ṅ =(n (V ) n) / τ n (V )
Comparison of HH models: 4D vs. 2D V 50 0 50 100 1 0, 8 0 50 100 Time m 1 0, 8 0, 6 0, 4 0, 2 1 0, 8 0 0 20 40 60 80 100 Time n 0, 6 0, 4 0, 2 0 0 20 40 60 80 100 Time h 0, 6 0, 4 0, 2 0 0 20 40 60 80 100 Time
Two-dimensional models Morris-Lecar model: INa,p+IK model: C V = I ḡ Ca m (V )(V E Ca ) ḡ K n(v E K ) g L (v ṅ = (n (V ) n)/τ n (V ) Fitzhugh-Nagumo model: C V = I ḡ Na m (V )(V E Na ) ḡ K n(v E K ) g L (V ṅ = (n (V ) n)/τ n (V ) ẋ = x(a x)(x 1) y + I ẏ = bx cy
Qualitative analysis of dynamical systems Example of 1D linear system: C dv dt = G(V V rest) = C V V 5 V > 0 V < 0 V V 0 V = f(v V rest ) 5 0 10 20 30 40 50 Time
Qualitative analysis of dynamical systems General 1D linear system: ẏ(t) = λy(t) Solution: y(t) = y 0 e λt ẏ ẏ ẏ ẏ > 0 ẏ < 0 y ẏ > 0 ẏ < 0 y ẏ < 0 ẏ > 0 y y 10 5 0 5 10 λ < 0 0 1 2 3 4 5 Time y 10 5 0 5 10 ẏ = λy 0 5 10 Time y 20 10 0 10 20 λ > 0 0 0, 5 1 Time
Qualitative analysis of dynamical systems Example of 1D nonlinear system: persistent sodium model. C V = I g L (V E L ) g Na m (V )(V V Na ) V V 50 0 50 0 50 V 0 50 0 2, 5 5 Time
Qualitative analysis of dynamical systems Example of 1D nonlinear system: persistent sodium model (I = 60) C V = I g L (V E L ) g Na m (V )(V V Na ) V 100 V 50 0 50 0 50 V 0 50 0 2, 5 5 Time
Qualitative analysis of dynamical systems 2D system:
INa,p+IK model: high threshold General behavior: I = 0
INa,p+IK model: high threshold General behavior: I = 0
INa,p+IK model: high threshold Saddle-node bifurcation: I = 4.75
INa,p+IK model: high threshold Saddle-node bifurcation: I = 6
INa,p+IK model: low threshold Supercritical Hopf bifurcation: I = 18.5
INa,p+IK model: low threshold Supercritical Hopf bifurcation: I = 21
INa,p+IK model: low threshold Supercritical Hopf bifurcation: I = 27
Hodgkin s classification (1948) Type I neurons: Type II neurons: Variable frequency Saddle-node / high threshold Constant frequency Hopf / low threshold
Bursting models (3D) Fast-slow system: Hindmarsh-Rose model: ẋ = y ax 3 + by 2 z + I ẏ = c dx 2 y ż = r[s(x x 0 ) z]
Hindmarsh-Rose model (1984) Two-dimensional: r = 0 Periodic behavior (limit cycle)
Hindmarsh-Rose model (1984) Three-dimensional: r = 0.006 Chaotic behavior (strange attractor)
Hybrid models Characteristics: Continuous part Discrete part Leaky integrate and fire (Lapicque, 1909) τ m V = V + RI (t) If V > V th then V V reset No spike generation mechanism
Quadratic integrate and fire τ V = V 2 + RI (t)
Two-dimensional hybrid model Izhikevich (2003)
Two-dimensional hybrid model
Simplified models Izhikevich (2004)
Discrete state automata Binary models: Excitable systems: Generic models: Quiescent Active Refractory
Models of synapses Synaptic weights: matrix representation Synaptic currents: Kinetic models Simplified models Learning rules:
Synaptic weights W = [w ij ] w ij > 0 for excitatory connections. w ij < 0 for inhibitory connections. With quantal release (n sites, p probability of release, q size of the quantum): w npq
Synaptic mechanisms
Transmitter release [T ](V pre ) = T max 1 + e ξ with ξ = (V pre v 1/2 ) K p x (V ) 1 2k 0.5 0 V 1/2 V
Synaptic currents: kinetic models I syn (t) = g syn (t)(v m (t) E syn ) with g syn (t) = ḡ syn s(t) Dependency to transmitter concentration Kinetics of current changes
Ionotropic receptors C + T α β O (1) with C for Closed and O for open. We get the equation: ds dt = α [T ] (1 s) βs (2) I AMPA = ḡ AMPA s(v m E AMPA ) (3) I GABAA = ḡ GABAA s(v m E GABAA ) (4) I NMDA = ḡ NMDA sm(v )(V E NMDA ) (5) and: M(V ) = 1 1 + exp( 0.062V ) [Mg2+] o /3.57 (6)
Metabotropic receptors R i + T K 1 R (7) K2 R + G i K3 R + G (8) G K4 G i (9) s n I GABAB = ḡ GABAB s n (V E K ) + K d with n = 4 (10) dr dt = K 1 [T ] (1 r) K 2 r (11) ds dt = K 3r K 4 s (12)
Synaptic currents: simplified models I syn (t) = g syn (t)(v m (t) E syn ) with g syn (t) = ḡ syn s(t) s(t) = k F (t t k ) with { F (t) = 0 for t < 0 F (t) > 0 otherwise s(t) The α function: F (t t k ) F (t) = f r f d f r f d (e f d t e fr t ) t 1 t 2 t 3 t 4 t 5 t