Reduction of the HH-Model to two dimensions (general)

Transcription

1 Spiking models 1

2 Neronal codes Spiking models: Hodgkin Hxley Model (brief repetition) Redction of the HH-Model to two dimensions (general) FitzHgh-Nagmo Model Integrate and Fire Model 2

3 Varying firing properties Rhythmic brst in the absence of synaptic inpts??? Inflence of steady hyperpolarization Inflence of the nerotransmitter Acetylcholin 3

4 Action Potential / Shapes: Sqid Giant Axon Rat - Mscle Cat - Heart 5 4

5 Yor nerons srely don t like this gy! 5

6 Voltage clamp method developed 1949 by Kenneth Cole sed in the 1950s by Alan Hodgkin and Andrew Hxley to measre ion crrent while maintaining specific membrane potentials 6

7 Voltage clamp method Large depolarization Small depolarization Ic: capacity crrent Il: leakage crrent 7

8 The sodim channel (patch clamp) 8

9 The sodim channel 9

10 Fnction of the sodim channel 10

11 Hodgkin and Hxley 11

12 12 Hodgkin Hxley Model: ) ( ) ( t I t I dv C inj k k m ) ( ) ( ) ( t I t I t I k k C inj with Q C and dv C d C I C ) ( ) ( ) ( 4 3 L m L K m K Na m Na k k V V g V V n g V V m h g I inj L m L K m K Na m Na m I V V g V V n g V V m h g dv C ) ( ) ( ) ( 4 3 charging crrent Ion channels ( x) m x x V V g I

13 13 Hodgkin Hxley Model: inj L m L K m K Na m Na m I V V g V V n g V V m h g dv C ) ( ) ( ) ( 4 3 h h h n n n m m m h h n n m m ) ( ) )(1 ( ) ( ) )(1 ( ) ( ) )(1 ( )] ( [ ) ( 1 x 0 x x x 1 0 )] ( ) ( [ ) ( )] ( ) ( [ ) ( x x x x x x x with voltage dependent gating variables time constant asymptotic vale

14 General redction of the Hodgkin-Hxley Model stimls C d g Na m 3 I Na h( V Na ) g K n I K 4 ( V K ) g l ( Ileak V L ) I( t) 1) dynamics of m are fast 2) dynamics of h and n are similar 14

15 General Redction of the Hodgkin-Hxley Model: 2 dimensional Neron Models stimls d F(, w) I( t) dw w G(, w) 15

16 FitzHgh-Nagmo Model d 3 3 w I dw ( w) dw 0.08( w) : membran potential w: recovery variable I: stimls 16

17 Introdction to dynamical systems Simple Example: Harmonic Oscillator 17

18 Introdction to dynamical systems Simple Example: Harmonic Oscillator Force is proportional to displacement of spring F = à kx = mẋ. 18

19 Introdction to dynamical systems Simple Example: Harmonic Oscillator Force is proportional to displacement of spring F = à kx = mẋ. Description as differential eqation: One second order eqation d 2x = à íx 2 19

20 Introdction to dynamical systems Simple Example: Harmonic Oscillator Force is proportional to displacement of spring.. F = à kx = mx Description as differential eqation: One second order eqation d 2x = à íx 2 Two first order eqations dx = v dv = à íx 20

21 Analysis of the Dynamics In the time domain: x and v have the same sinsoidal time corse bt have a phase shift of π/2, i.e. when x is in eqillibrim position velocity is maximal and when velocity is zero x reaches the amplitde. x v t xç =v vç =à íx 21

22 Analysis of the Dynamics In the time domain: x and v have the same sinsoidal time corse bt have a phase shift of π/2, i.e. when x is in eqillibrim position velocity is maximal and when velocity is zero x reaches the amplitde. x v t xç =v vç =à íx Phase space: Abstract representation of all dynamical parameters v ax 2 + bv 2 = constant Energy defines shape => Ellipse x E = E kin + E pot = 2 1 mv kx 2 = constant 22

23 Analysis of the Dynamics In the time domain: x and v have the same sinsoidal time corse bt have a phase shift of π/2, i.e. when x is in eqillibrim position velocity is maximal and when velocity is zero x reaches the amplitde. x v t xç =v vç =à íx Phase space: Abstract representation of all dynamical parameters v x Points of retrn: v = 0 23

24 Analysis of the Dynamics In the time domain: x and v have the same sinsoidal time corse bt have a phase shift of π/2, i.e. when x is in eqillibrim position velocity is maximal and when velocity is zero x reaches the amplitde. x v t xç =v vç =à íx Phase space: Abstract representation of all dynamical parameters v Eqilibrim Position x=0: v = v max and v = -v max x 24

25 Analysis of the Dynamics In the time domain: x and v have the same sinsoidal time corse bt have a phase shift of π/2, i.e. when x is in eqillibrim position velocity is maximal and when velocity is zero x reaches the amplitde. x v t xç =v vç =à íx Phase space: Abstract representation of all dynamical parameters v Eqilibrim Position x=0: v = v max and v = -v max!!! Important: Same position x = 0, bt two different velocities. Happens often that observable (position) has one vale, bt system is in different STATES (moving p or moving down). x 25

26 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point 26

27 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point v Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) x. Arrow p (or down) means x=0, velocity at the location (x,v) does not change AND. v=large (in this case even maximal). This is evident for these two locations as they are the reversal points of the pendlm, where there is no velocity bt a maximal acceleration. 27

28 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point v Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0) x 28

29 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point v Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0)... and others. x 29

30 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point v Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0) x... and others. This is special and idealized case: Trajectory in phase space (ellipse) remains the same. 30

31 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point v Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0) x... and others. This is special and idealized case: Trajectory in phase space (ellipse) remains the same. It depends only on the the initial conditions (how far we strain the spring in the beginning). 31

32 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0)... and others. A denser pictre of a circlar vector field. 32

33 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0)... and others. Here is an example for an oscillator with damping: The phase space trajectory always retrns to the resting position and (x, v) =(0, 0) (xç,vç) = (0, 0) 33

34 Vector Field We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0)... and others. And vector fields can be really crazy... 34

35 Fixed Points For analysis of dynamical systems there are points of special interest where the system or a particlar variable does not change, i.e. the vector (xç,vç) = (0, 0) 35

36 Fixed Points For analysis of dynamical systems there are points of special interest where the system or a particlar variable does not change, i.e. the vector (xç,vç) = (0, 0) Look at or simple example: xç =v vç =à íx For which vales (x,v) is the pper eqation tre and what does this soltion mean? 36

37 Fixed Points For analysis of dynamical systems there are points of special interest where the system or a particlar variable does not change, i.e. the vector (xç,vç) = (0, 0) Look at or simple example: xç =v vç =à íx For which vales (x,v) is the pper eqation tre and what does this soltion mean? Only for one single point (x, v) =(0, 0), the FIXED POINT 37

38 Fixed Points For analysis of dynamical systems there are points of special interest where the system or a particlar variable does not change, i.e. the vector (xç,vç) = (0, 0) Look at or simple example: xç =v vç =à íx For which vales (x,v) is the pper eqation tre and what does this soltion mean? Only for one single point (x, v) =(0, 0), the FIXED POINT In the simplest case: a resting pendlm! This is also tre for the damped oscillator, bt there the fixed point is also an ATTRACTOR, i.e. no matter where we start in phase space we will always end p at the resting position. No srprise! 38

39 Fixed Points II and Other Stff Consider a more general case: xç =f(x, v) vç =g(x, v) Still interesting where the system doesn't change: xç =f(x, v) =0 vç =g(x, v) =0 39

40 Fixed Points II and Other Stff Consider a more general case: xç =f(x, v) vç =g(x, v) Still interesting where the system doesn't change: The geometric fnctions that we get from these eqations are called NULLCLINES. xç =f(x, v) =0 vç =g(x, v) =0 They signify where the vectors of the vector fields are only horizontal or vertical. At the intersection of the the nllclines we find fixed points. 40

41 Fixed Points II and Other Stff Consider a more general case: xç =f(x, v) vç =g(x, v) Still interesting where the system doesn't change: The geometric fnctions that we get from these eqations are called NULLCLINES. xç =f(x, v) =0 vç =g(x, v) =0 They signify where the vectors of the vector fields are only horizontal or vertical. At the intersection of the the nllclines we find fixed points. For the simple spring example we find the trivial nllclines: They intersect at (x, v) =(0, 0) xç =0 v(x) =0 vç =0 x(v) =0 which is the fixed point. 41

42 Nllclines We have for variables: x, v, xç,vç xç =v vç =à íx The change of a point in phase space can be depicted by drawing (xç,vç) (x, v) the vector of change at each point v Examples: (x, v) =(æ x max, 0) (xç,vç) = (0, ç íx) (x, v) =(0, æ v max ) (xç,vç) = (æ v max, 0) x 42

43 Limit cycles We know already a simple example of an attractor: A fixed point (red cross). 43

44 Limit cycles In more complex systems we find other kinds of attractors sch as LIMIT CYCLES Attracting circlar trajectories approached by all other trajectories no matter whether they start from the otside (ble spirals) or from the inside (green spirals). 44

45 Limit cycles In more complex systems we find other kinds of attractors sch as LIMIT CYCLES Attracting circlar trajectories approached by all other trajectories no matter whether they start from the otside (ble spirals) or from the inside (green spirals). Attracting trajectory is called limit cycle becase mathematically it is reached for all other trajectories in the limit of infinite time they are getting closer and closer bt in theory never qite reach the limit cycle. 45

46 Limit cycles II Another famos example is the Van-der-Pol-oscillator: The shape of the limit cycle depends on parameters in the differential eqations. 46

47 General Redction of the Hodgkin-Hxley Model: 2 dimensional Neron Models C dv m g Na 3 m h( V m V Na ) g K n 4 ( V m V K ) g L ( V m V L ) I inj stimls d F(, w) I( t) dw w G(, w) 47

48 Phase space = + w w = - w = w 0 0 w = -w 48

49 49 Hodgkin Hxley Model: inj L m L K m K Na m Na m I V V g V V n g V V m h g dv C ) ( ) ( ) ( 4 3 h h h n n n m m m h h n n m m ) ( ) )(1 ( ) ( ) )(1 ( ) ( ) )(1 ( voltage dependent gating variables m -> 0 h = const. Assmptions that lead to the FH-Model

50 FitzHgh-Nagmo Model d dw dw 3 3 w I ( w) 0.08( w) : membran potential w: recovery variable I: stimls 50

51 FitzHgh-Nagmo Model d dw 3 3 w I ( w) nllclines d dw

52 The vector field always points towards the stable fixed points! Ths, at every stable fixed point vectors mst trn arond. w Ball in a bowl Ball on a bowl dw 0 Stable fixpoint (vectors point towards) Unstable fixpoint (vectors point away) d 0 52

53 The vector direction at a nllcline is always either horizontal (ble) or vertical (red). Some starting point w Ths, we can often approximate the behavior of the system by treating the whole vector field as being either horizontal or vertical. From every given point we jst follow the initial vector (which we mst know) ntil we hit a nllcline, then trn as appropriate and move on to the next nllcline, trn 90 deg now and so on! etc. dw 0 d 0 53

54 FitzHgh-Nagmo Model nllclines d 3 3 stimls w I w dw 0 Adding a constat term to an eqation shifts the crve p- (or down-)wards! I(t)=I 0 dw ( w) d 54 0

55 FitzHgh-Nagmo Model w We receive a contraction to a new fixed point! d dw nllclines 3 3 w I ( w) The system got shifted a bit by crrent inpt I(t)=I 0 d 0 dw 0 We had been here (and stable) We get a new stable fixed point as soon as the minimm of the new red nllcline is lower and to the right of the old fixed point. 55

56 FitzHgh-Nagmo Model We wold receive an expansion (divergence) in this case. d nllclines 3 3 w I w d 0 p, p! The system got shifted a lot by crrent inpt dw ( w) I(t)=I 0 dw 0 56 We had been here (and stable)

57 FitzHgh-Nagmo Model nllclines Things are not that bad! As the vector field has a crl it will not diverge. Rather we get a limit cycle! d 3 3 stimls w I w dw 0 dw ( w) I(t)=I 0 d 0 limit cycle represents spiking 57

58 FitzHgh-Nagmo Model 58

59 FitzHgh-Nagmo Model Green area: Passive repolarization, no spike! 59

60 FitzHgh-Nagmo Model Green area: Active process, spike! 60

61 The FitzHgh-Nagmo model Absence of all-or-none spikes (java applet) no well-defined firing threshold weak stimli reslt in small trajectories ( sbthreshold response ) strong stimli reslt in large trajectories ( sprathreshold response ) BUT: it is only a qasi-threshold along the nstable middle branch of the V-nllcline 61

62 The Fitzhgh-Nagmo model Anodal break excitation Post-inhibitory (rebond) spiking: transient spike after hyperpolarization V Original threshold Shows the effect of the qasi-threshold! New threshold t 62

63 Being absolte refractory Even with a lot of crrent we will always hit the left branch of the red (now green!) nllcline and then travel done again. No new spike can be elicted! 63

64 Being relative refractory With a little crrent (green) we will again hit the downward branch of the nllcline (as before) with a lot of crrent (grey) we will, however, hit the ble (other) nllcline and a etc. new spike will be elicited! etc. 64

65 The FitzHgh-Nagmo model Excitation block and periodic spiking Increasing I shifts the V-nllcline pward -> periodic spiking as long as eqilibrim is on the nstable middle branch -> Oscillations can be blocked (by excitation) when I increases frther 65

66 The Fitzhgh-Nagmo model Spike accommodation no spikes when slowly depolarized transient spikes at fast depolarization 66

67 Neronal codes Spiking models: Hodgkin Hxley Model (brief repetition) Redction of the HH-Model to two dimensions (general) FitzHgh-Nagmo Model Integrate and Fire Model 67

68 Integrate and Fire model j i Spike emission i Spike reception models two key aspects of neronal excitability: passive integrating response for small inpts stereotype implse, once the inpt exceeds a particlar amplitde 68

69 Integrate and Fire model j i Spike emission I i Spike reception: EPSP reset m d t ( t) RI( t) i Fire+reset threshold 69

70 Integrate and Fire model I(t) Time-dependent inpt i I(t) -spikes are events -threshold -spike/reset/refractoriness 70

71 Integrate and Fire model (linear) d ( t) RI( t) If firing: 0 d I=0 d I>0 0 resting t repetitive 71 t

72 Integrate and Fire model d ( t) RI( t) d F( ) RI( t) linear non-linear If firing: 0 72

73 Integrate and Fire model (non-linear) d I=0 d I>0 d t F( ) RI( t) Fire+reset non-linear threshold F Qadratic I&F: 2 ( ) c2 c0 73

74 Integrate and Fire model (non-linear) d I=0 d I>0 d t F( ) RI( t) Fire+reset non-linear threshold F Qadratic I&F: 2 ( ) c2 c0 exponential I&F: F( ) c0 exp( ) 74

75 Integrate and Fire model (non-linear) d I=0 critical voltage for spike initiation 75

76 Linear integrate-and-fire: Strict voltage threshold - by constrction - spike threshold = reset condition Non-linear integrate-and-fire: There is no strict firing threshold - firing depends on inpt - exact reset condition of minor relevance 76

77 Comparison: detailed vs non-linear I&F I C g Na g Kv1 g Kv3 g l d I(t) d F( ) RI( t) 77

78 neronal featres 78

79 79 E.M. Izhikevich, IEEE Transactions on Neral Networks, 15: , 2004

80 E.M. Izhikevich, w I w a ( b w) if 30mV threshold c w w d 80

81 w w a ( b w) 30mV c w w w d I 81

82 End here 82

83 Neronal codes Spiking models: Hodgkin Hxley Model (small regeneration) Redction of the HH-Model to two dimensions (general) FitzHgh-Nagmo Model Integrate and Fire Model Spike Response Model 83

84 Spike response model (for details see Gerstner and Kistler, 2002) = generalization of the I&F model SRM: parameters depend on the time since the last otpt spike integral over the past I&F: voltage dependent parameters differential eqations allows to model refractoriness as a combination of three components: 1. redced responsiveness after an otpt spike 2. increase in threshold after firing 3. hyperpolarizing spike after-potential 84

85 Spike response model (for details see Gerstner and Kistler, 2002) j i Spike reception: EPSP f t t j Spike emission: AP i Spike reception: EPSP f t t j Spike emission ^ t t i ^ t t i Last spike of i form of the AP and the after-potential t i t t t ^ i j f ^ i Firing: t t i w ij t t synaptic efficacy All spikes, all nerons f j time corse of the response to an incoming spike 85

86 Spike response model (for details see Gerstner and Kistler, 2002) j i i i t t t ^ i 0 j f w ij t t ^ k( t t, s) I ( t s) ds i ext f j external driving crrent 86

87 Spike response model dynamic threshold ( t t ' ) ^ t i threshold t d( t) 0 Firing: t' t 87

88 Comparison: detailed vs SRM <2ms 80% of spikes correct (+/-2ms) C I(t) g Na I g Kv1 g Kv3 g l Spike detailed model threshold model (SRM) 88

89 References Rieke, F. et al. (1996). Spikes: Exploring the neral code. MIT Press. Izhikevich E. M. (2007) Dynamical Systems in Neroscience: The Geometry of Excitability and Brsting. MIT Press. Fitzhgh R. (1961) Implses and physiological states in theoretical models of nerve membrane. Biophysical J. 1: Nagmo J. et al. (1962) An active plse transmission line simlating nerve axon. Proc IRE. 50: Gerstner, W. and Kistler, W. M. (2002) Spiking Neron Models. Cambridge University Press. online at: 89