Single neuron models. L. Pezard Aix-Marseille University

Transcription

1 Single neuron models L. Pezard Aix-Marseille University

2 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

3 Biological neuron

4 The membrane Models of the electrophysiology of excitability

5 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

6 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 )

7 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 )

8 Ionic distributions Cell of Mammal (T = 37 o C): Gradients maintenance: Ions c i c e E k K mv Na + from 5 to to 90 mv Cl 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

9 Ionic currents Ionic channels: Models: Characteristic: Non-gated channels Gated channels Electrodiffusion Barriers models State models (kinetics and stochastics) I-V curve

10 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

11 I-V curves for GHK model I(a.u.) c o /c i = 1 c o /c i = 5 c o /c i = 10 c o /c i = V (mv )

12 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

13 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

14 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 )

15 Cable equation (Rall, ) 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

16 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

17 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

18 Sodium gated channel

19 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

20 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

21 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

22 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 )

23 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 ) = 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)

24 Potential dependence of kinetics parameters Time constant Asymptotic value 8 1 τ (ms) τ m τ h τ n w 0, 8 0, 6 0, 4 0, 2 m h n V (mv) V (mv) τ x (V ) = 1 α x (V ) + β x (V ) x (V ) = α x (V ) α x (V ) + β x (V )

25 Hodgkin-Huxley model 50 V Time , 8 P 0, 6 0, 4 h n G g Na g K 0, Time m Time

26 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 )

27 Standard kinetic parameters x (V ) 1 2k τ x (V ) C b + C a 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

28 Zoology of ion channels

29 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

30 Compartmental models Principle:

31 Compartmental models Simulation environments: NEURON, GENESIS

32 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)

33 Reduction of Hodgkin-Huxley model: step 1 Separation of time scales: m = m (V ) τ (ms) τ m τ h τ n P m n h 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 )

34 Comparison of HH models: 4D vs. 3D , 8 V , 6 P 0, 4 0, Time Time 1 0, 8 0, 8 P 0, 6 0, 4 P 0, 6 0, 4 0, 2 0, Time Time

35 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: Time h 0, 75 0, 5 0, , 8 h = n 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 )

36 Comparison of HH models: 4D vs. 2D V , Time m 1 0, 8 0, 6 0, 4 0, 2 1 0, Time n 0, 6 0, 4 0, Time h 0, 6 0, 4 0, Time

37 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

38 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 ) Time

39 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 λ < Time y ẏ = λy Time y λ > 0 0 0, 5 1 Time

40 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 V , 5 5 Time

41 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 V , 5 5 Time

42 Qualitative analysis of dynamical systems 2D system:

43 INa,p+IK model: high threshold General behavior: I = 0

44 INa,p+IK model: high threshold General behavior: I = 0

45 INa,p+IK model: high threshold Saddle-node bifurcation: I = 4.75

46 INa,p+IK model: high threshold Saddle-node bifurcation: I = 6

47 INa,p+IK model: low threshold Supercritical Hopf bifurcation: I = 18.5

48 INa,p+IK model: low threshold Supercritical Hopf bifurcation: I = 21

49 INa,p+IK model: low threshold Supercritical Hopf bifurcation: I = 27

50 Hodgkin s classification (1948) Type I neurons: Type II neurons: Variable frequency Saddle-node / high threshold Constant frequency Hopf / low threshold

51 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]

52 Hindmarsh-Rose model (1984) Two-dimensional: r = 0 Periodic behavior (limit cycle)

53 Hindmarsh-Rose model (1984) Three-dimensional: r = Chaotic behavior (strange attractor)

54 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

55 Quadratic integrate and fire τ V = V 2 + RI (t)

56 Two-dimensional hybrid model Izhikevich (2003)

57 Two-dimensional hybrid model

58 Simplified models Izhikevich (2004)

59 Discrete state automata Binary models: Excitable systems: Generic models: Quiescent Active Refractory

60 Models of synapses Synaptic weights: matrix representation Synaptic currents: Kinetic models Simplified models Learning rules:

61 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

62 Synaptic mechanisms

63 Transmitter release [T ](V pre ) = T max 1 + e ξ with ξ = (V pre v 1/2 ) K p x (V ) 1 2k V 1/2 V

64 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

65 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 ) = exp( 0.062V ) [Mg2+] o /3.57 (6)

66 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)

67 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