Neuronal Dynamics: Computational Neuroscience of Single Neurons

Transcription

1 Week 4 part 1 : Redction of the Hodgkin-Hxley Model Neronal Dynamics: Comptational Neroscience of Single Nerons Week 4 Redcing detail: Two-dimensional neron models Wlfram Gerstner EPFL, Lasanne, Switzerland 4.1 From Hodgkin-Hxley to 2D - Overview: From 4 to 2 dimensions - MathDetor 1: Separation of time scales - MathDetor 2: Exploiting similarities 4.2 Phase Plane Analysis - Role of nllclines 4.3 Analysis of a 2D Neron Model - MathDetor 3: Stability of fixed points 4.4 Type and Neron Models - where is the firing threshold? - separation of time scales 4.5. Nonlinear ntegrate-and-fire - from two to one dimension

2 Week 4 part 1 : Redction of the Hodgkin-Hxley Model 4.1 From Hodgkin-Hxley to 2D - Overview: From 4 to 2 dimensions - MathDetor 1: Separation of time scales - MathDetor 2: Exploiting similarities 4.2 Phase Plane Analysis - Role of nllclines 4.3 Analysis of a 2D Neron Model - MathDetor 3: Stability of fixed points 4.4 Type and Neron Models - where is the firing threshold? - separation of time scales 4.5. Nonlinear ntegrate-and-fire - from two to one dimension

3 Neronal Dynamics 4.1. Review :Hodgkin-Hxley Model t () cortical neron d τ =... à Hodgkin-Hxley model à Compartmental models

4 Neronal Dynamics 4.1 Review :Hodgkin-Hxley Model Week 2: Cell membrane contains - ion channels - ion pmps -70mV K + Na + Ca 2+ ons/proteins Dendrites (week 3): Active processes? assmption: (mainly) passive à point neron action potential

5 Neronal Dynamics 4.1. Review :Hodgkin-Hxley Model 100 mv 0 on channels on pmp Reversal potential kt inside Ka Na otside ln n ( 1) 1 2 n ( ) Δ = = ion pmps à concentration difference ó voltage difference q K 2

6 Neronal Dynamics 4.1. Review: Hodgkin-Hxley Model C C d 100 mv g K 0 3 Na = g Na m h( ENa) gk n ( EK ) gl ( El ) + dh dm dn hn m h nm ( 0 0 ) ) ( 0 ) = ( () ) τ h n τ m g Na Hodgkin and Hxley, 1952 g l K 4 on channels stimls leak n 0 () ( t) on pmp τ n () inside Ka Na otside 4 eqations = 4D system

7 Neronal Dynamics 4.1. Overview and aims Can we nderstand the dynamics of the HH model? - mathematical principle of Action Potential generation? - Types of neron model (type and )? - threshold behavior? à Redce from 4 to 2 eqations ramp inpt/ constant inpt f Type and f- crve type models f f- crve 0 0 0

8 Neronal Dynamics 4.1. Overview and aims Toward a two-dimensional neron model - Redction of Hodgkin-Hxley to 2 dimension -step 1: separation of time scales -step 2: exploit similarities/correlations

9 Neronal Dynamics 4.1. Redction of Hodgkin-Hxley model K leak 3 4 Na Na K K l l = + Na d C g m h( E ) g n ( E ) g ( E ) ( t) dm dh dn m m ( 0 ) = ( ) τ m h h ( 0 ) = τ h ( ) n n ( 0 ) = ( ) τ n m 0 () τ h () h 0 () τ m () stimls MathDetor 4.1 1) dynamics of m is fast m ( t) = m0 ( ( t))

10 Neronal Dynamics 4.1. Redction of Hodgkin-Hxley model stimls K leak 3 4 Na Na K K l l = + Na d C g m h( E ) g n ( E ) g ( E ) ( t) dm dh dn m m ( 0 ) = ( ) τ m h h ( 0 ) = τ h ( ) n n ( 0 ) = ( ) τ n 1) dynamics of m are fast 2) dynamics of h and n are similar n 0 () h 0 () τ n τ h () τ m () m ( t) = m0 ( ( t))

11 Neronal Dynamics 4.1. Redction of Hodgkin-Hxley model stimls d C g m h( E ) g n ( E ) g ( E ) ( t) 3 4 Na Na K K l l = + 2) dynamics of h and n are similar 1 h ( t) = a n( t) MathDetor 4.2 t 1 hrest nrest 0 h(t) n(t) t

12 Neronal Dynamics 4.1. Redction of Hodgkin-Hxley model Na K leak d C g m t h t t E g n t t E g t E t = Na[ ( )] ( )( ( ) Na) K[ ( )] ( ( ) K ) l ( ( ) l ) ( ) d w C = g m w E g a E g E + t m ( t) = m0 ( ( t 1 h ( t) = a n( t) 3 4 Na 0 ( ) (1 )( Na) K[ ] ( K ) l ( l ) ( ) 1) dynamics of m are fast )) 2) dynamics of h and n are similar w(t) w(t)

13 Neronal Dynamics 4.1. Redction of Hodgkin-Hxley model Na K leak d w C = g m w E g E g E + t a dw 3 4 Na 0 ( ) (1 )( Na) K ( ) ( K ) l ( l ) ( ) w w ( ) 0 = τ ( ) eff d C = f( ( t), w( t)) + ( t) dw = gt ( ( ), wt ( ))

14 Neronal Dynamics 4.1. Redction to 2 dimensions 2-dimensional eqation d C = f( ( t), w( t)) + ( t) dw = gt ( ( ), wt ( )) Enables graphical analysis! -Discssion of threshold -Type and - Repetitive firing

15 Neronal Dynamics Qiz 4.1. A- Assmptions: n order to redce a detailed compartmental neron model to two dimensions we have to assme that [ ] dendrites can be approximated as passive [ ] the neron model has no dendrite [ ] the neron model has at most 2 types of ion channels [ ] all gating variables are fast [ ] no gating variable is fast [ ] gating variables fall in two grops: those that are fast and those that are slow [ ] at least one of the ion channels is inactivating [ ] the neron does not generate spikes B - A biophysical point model with 3 ion channels, each with activation and inactivation, has a total nmber of eqations eqal to [ ] 3 or [ ] 4 or [ ] 6 or [ ] 7 ; [ ] 8 or more C- Separation of time scales: We start with two eqations dx τ1 = x+ () t dy 2 τ 2 = y+ x + A We assme that τ = 1 2 n this case a redction of dimensionality [ ] is not possible [ ] is possible and the reslt is dy 2 τ 2 = y+ [()] t + A [ ] is possible and the reslt is dx 2 τ 1 = x+ x + A τ