Biological Modeling of Neural Networks
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1 Week 3 part 1 : Rection of the Hodgkin-Huxley Model 3.1 From Hodgkin-Huxley to 2D Biological Modeling of Neural Netorks - Overvie: From 4 to 2 dimensions - MathDetour 1: Exploiting similarities - MathDetour 2: Separation of time scales 3.2 Phase Plane Analysis - Role of nullclines Week 3 Recing detail: To-dimensional neuron models Wulfram Gerstner EPFL, Lausanne, Sitzerland 3.3 Analysis of a 2D Neuron Model - constant input vs pulse input - MathDetour 3: Stability of fixed points 3.4 TypeI and II Neuron Models next eek!
2 Neuronal Dynamics 3.1. Revie :Hodgkin-Huxley Model cortical neuron It () u d u... Hodgkin-Huxley model Compartmental models
3 Neuronal Dynamics 3.1 Revie :Hodgkin-Huxley Model Week 2: Cell membrane contains - ion channels - ion pumps -70mV Dendrites (eek 4): Active processes? assumption: passive dendrite point neuron Na + spike generation action potential K + Ca 2+ Ions/proteins
4 Neuronal Dynamics 3.1. Revie :Hodgkin-Huxley Model 100 inside K Ka mv 0 Ion channels Ion pump Na outside ln nu ( ) nu ( ) u u u Reversal potential kt ion pumps concentration difference voltage difference q 2
5 Neuronal Dynamics 3.1. Revie: Hodgkin-Huxley Model 100 I Hodgkin and Huxley, 1952 inside Ka C mv g K 0 g Na g l Ion channels Ion pump Na outside stimulus I Na I K I leak 4 equations C g Na m 3 h( u E Na ) g K n 4 ( u E K ) g l ( u E l ) I ( t) = 4D system dh dm dn hn m h n0( m u0( ) u) ( u () u ) hn m n 0 (u) n (u) u u
6 Neuronal Dynamics 3.1. Overvie and aims Can e understand the dynamics of the HH model? - mathematical principle of Action Potential generation? - constant input current vs pulse input? - Types of neuron model (type I and II)? (next eek) - threshold behavior? (next eek) Rece from 4 to 2 equations Type I and type II models ramp input/ constant input f f-i curve f f-i curve I0 I0 I0
7 Neuronal Dynamics 3.1. Overvie and aims Can e understand the dynamics of the HH model? Rece from 4 to 2 equations
8 Neuronal Dynamics Quiz 3.1. A - A biophysical point neuron model ith 3 ion channels, each ith activation and inactivation, has a total number of equations equal to [ ] 3 or [ ] 4 or [ ] 6 or [ ] 7 or [ ] 8 or more
9 Neuronal Dynamics 3.1. Overvie and aims Toard a to-dimensional neuron model -Rection of Hodgkin-Huxley to 2 dimension -step 1: separation of time scales -step 2: exploit similarities/correlations
10 Neuronal Dynamics 3.1. Rection of Hodgkin-Huxley model stimulus I I I Na K leak C g m 3 h( u E ) g n 4 ( u E ) g ( u E ) I ( t) Na Na K K l l dm m m0( u) m 0 (u) (u) h m ( u) dh h h0( u) h 0 (u) m (u) h ( u) u u dn n n0( u) n ( u) MathDetour 1) dynamics of m are fast m( t) m ( u( t)) 0
11 Neuronal Dynamics MathDetour 3.1: Separation of time scales dx x c() t 1 To coupled differential equations 1 dx x h( y) dy Exercise 1 (eek 3) 2 f ( y) g( x) (later!) Separation of time scales x h( y) 1 2 Reced 1-dimensional system dy f ( y) g( h( y)) 2
12 Neuronal Dynamics 3.1. Rection of Hodgkin-Huxley model stimulus I I I Na K leak C g m 3 h( u E ) g n 4 ( u E ) g ( u E ) I ( t) Na Na K K l l dm m m0( u) m ( u) n 0 (u) n h (u) dh h h0( u) h ( u) h 0 (u) u m (u) u dn n n0( u) n ( u) 1) dynamics of m are fast m( t) m ( u( t)) 0 2) dynamics of h and n are similar
13 Neuronal Dynamics 3.1. Rection of Hodgkin-Huxley model Rection of Hodgkin-Huxley Model to 2 Dimension -step 1: separation of time scales -step 2: exploit similarities/correlations No!
14 Neuronal Dynamics 3.1. Rection of Hodgkin-Huxley model stimulus C g m 3 h( u E ) g n 4 ( u E ) g ( u E ) I ( t) Na Na K K l l 2) dynamics of h and n are similar 1 h( t) a n ( t) u MathDetour t 1 hrest nrest 0 h(t) n(t) t
15 Neuronal Dynamics Detour 3.1. Exploit similarities/correlations dynamics of h and n are similar h 1 h( t) a n( t ) 1-h Math. argument n 1 hrest nrest 0 h(t) n(t) t
16 Neuronal Dynamics Detour 3.1. Exploit similarities/correlations dynamics of h and n are similar 1 h( t) a n( t ) at rest n dh h h0( u) h ( u) 1 hrest nrest h(t) n(t) dn n n0( u) n ( u) 0 t
17 Neuronal Dynamics Detour 3.1. Exploit similarities/correlations dynamics of h and n are similar (i) Rotate coordinate system (ii) Suppress one coordinate (iii) Express dynamics in ne variable 1 h( t) a n( t) ( t) 1 hrest nrest n h(t) n(t) dh dn h h0( u) h n n0( u) n ( u) ( u) d 0 ( u ) eff ( u) 0 t
18 Neuronal Dynamics 3.1. Rection of Hodgkin-Huxley model I I Na K C g [ m( t)] 3 h( t)( u( t) E ) g [ n( t)] 4 ( u( t) E ) g ( u( t) E ) I( t) Na Na K K l l Ileak C g m ( u) 3 (1 )( u E ) g [ ] 4 ( u E ) g ( u E ) I( t) Na 0 Na K K l l a 1) dynamics of m are fast m( t) m ( u( t)) 0 2) dynamics of h and n are similar 1 h( t) a n( t) (t) (t) dh h h0( u) h ( u) d 0 ( u ) dn n n0( u) eff ( u) n ( u)
19 Neuronal Dynamics 3.1. Rection of Hodgkin-Huxley model I Na I K Ileak C g m ( u) 3 (1 )( u E ) g ( ) 4 ( u E ) g ( u E ) I( t) Na 0 Na K K l l a d 0 ( u ) eff ( u) F( u( t), ( t)) R I( t) d G ( u ( t ), ( t ))
20 NOW Exercise : separation of time scales C g m 3 h( u E ) g n 4 ( u E ) g ( u E ) I ( t) Na Na K K l l dm Exercises: m m0( u) m ( u) A: dx x c(t) - calculate x(t)! x(t) c(t) 0 1 t no! - hat if is small?? 1.5 homeork Exerc. 9h50-10h00 Next lecture: 10h15 dm m c( u) m f ( u) m B: -calculate m(t) if is small! - rece to 1 eq. t
21 Week 3 part 1 : Rection of the Hodgkin-Huxley Model 3.1 From Hodgkin-Huxley to 2D Biological Modeling of Neural Netorks - Overvie: From 4 to 2 dimensions - MathDetour 1: Exploiting similarities - MathDetour 2: Separation of time scales 3.2 Phase Plane Analysis - Role of nullclines Week 3 Recing detail: To-dimensional neuron models Wulfram Gerstner EPFL, Lausanne, Sitzerland 3.3 Analysis of a 2D Neuron Model - constant input vs pulse input - MathDetour 3: Stability of fixed points 3.4 TypeI and II Neuron Models next eek!
22 Neuronal Dynamics MathDetour 3.1: Separation of time scales To coupled differential equations dx Ex. 1-A x c() t 1 1 dx x c() t dy Exercise 1 (eek 3) 2 f ( y) g( x) Separation of time scales 1 2 Reced 1-dimensional system dy f ( y) g( c( t)) 2
23 Neuronal Dynamics MathDetour 3.2: Separation of time scales Linear differential equation dx x c() t 1 step
24 Neuronal Dynamics MathDetour 3.2 Separation of time scales To coupled differential equations 1 dx x c() t x 2 dc c f ( x) I( t) 1 2 c slo drive I
25 Neuronal Dynamics Rection of Hodgkin-Huxley model stimulus I I I Na K leak C g m 3 h( u E ) g n 4 ( u E ) g ( u E ) I ( t) Na Na K K l l dm m m0( u) m ( u) n 0 (u) n h (u) dh h h0( u) h ( u) h 0 (u) u m (u) u dn n n0( u) n ( u) dynamics of m is fast m( t) m ( u( t)) 0 Fast compared to hat?
26 Neuronal Dynamics MathDetour 3.2: Separation of time scales To coupled differential equations dx 1 x h( y) dy Exercise 1 (eek 3) 2 f ( y) g( x) Separation of time scales even more general x h( y) 1 2 Reced 1-dimensional system dy f ( y) g( h( y)) 2
27 Neuronal Dynamics Quiz 3.2. A- Separation of time scales: We start ith to equations dx x y I() t 1 2 dy [ ] If then the system can be reded to 2 y x A 1 2 dy y [ y I( t)] 2 A 2 Attention I(t) can move rapidly, therefore choice [1] not correct 2 1 [ ] If then the system can be reded to dx x x 2 A I() t 1 [ ] None of the above is correct.
28 Neuronal Dynamics Quiz 3.2-similar dynamics Exploiting similarities: A sufficient condition to replace to gating variables r,s by a single gating variable is [ ] Both r and s have the same time constant (as a function of u) [ ] Both r and s have the same activation function [ ] Both r and s have the same time constant (as a function of u) AND the same activation function [ ] Both r and s have the same time constant (as a function of u) AND activation functions that are identical after some additive rescaling [ ] Both r and s have the same time constant (as a function of u) AND activation functions that are identical after some multiplicative rescaling
29 Neuronal Dynamics 3.1. Rection to 2 dimensions 2-dimensional equation C f ( u( t), ( t)) I( t) d g ( u ( t ), ( t )) Enables graphical analysis! Phase plane analysis -Discussion of threshold - Constant input current vs pulse input -Type I and II - Repetitive firing
30 Week 3 part 1 : Rection of the Hodgkin-Huxley Model 3.1 From Hodgkin-Huxley to 2D - Overvie: From 4 to 2 dimensions Biological Modeling of Neural Netorks - MathDetour 1: Exploiting similarities - MathDetour 2: Separation of time scales 3.2 Phase Plane Analysis - Role of nullclines 3.3 Analysis of a 2D Neuron Model Week 3 Recing detail: To-dimensional neuron models Wulfram Gerstner EPFL, Lausanne, Sitzerland - constant input vs pulse input - MathDetour 3: Stability of fixed points 3.4 TypeI and II Neuron Models next eek!
31 Neuronal Dynamics 3.2. Reced Hodgkin-Huxley model I Na I K Ileak C g m ( u) 3 (1 )( u E ) g ( ) 4 ( u E ) g ( u E ) I( t) Na 0 Na K K l l a d 0 ( ( u) u) stimulus F( u, ) RI( t) d G( u, )
32 Neuronal Dynamics 3.2. Phase Plane Analysis/nullclines 2-dimensional equation stimulus F( u, ) RI( t) d G( u, ) u-nullcline Enables graphical analysis! -Discussion of threshold -Type I and II -nullcline
33 Neuronal Dynamics 3.2. FitzHugh-Nagumo Model F( u, ) RI ( t) 1 u u 3 RI () t 3 MathAnalysis, blackboard d G ( u, ) b b u 0 1 u-nullcline -nullcline
34 Neuronal Dynamics 3.2. flo arros F( u, ) RI( t) Stimulus I=0 d 0 d G( u, ) Consider change in small time step Flo on nullcline I(t)=0 u Flo in regions beteen nullclines Stable fixed point 0
35 Neuronal Dynamics Quiz 3.3 A. u-nullclines [ ] On the u-nullcline, arros are alays vertical [ ] On the u-nullcline, arros point alays vertically upard [ ] On the u-nullcline, arros are alays horizontal [ ] On the u-nullcline, arros point alays to the left [ ] On the u-nullcline, arros point alays to the right Take 1 minute, continue at 10:55 B. -Nullclines [ ] On the -nullcline, arros are alays vertical [ ] On the -nullcline, arros point alays vertically upard [ ] On the -nullcline, arros are alays horizontal [ ] On the -nullcline, arros point alays to the left [ ] On the -nullcline, arros point alays to the right [ ] On the -nullcline, arros can point in an arbitrary direction
36 Neuronal Dynamics 4.2. FitzHugh-Nagumo Model F( u, ) RI ( t) 1 u u 3 RI () t 3 d G ( u, ) b b u 0 1 change b1
37 Neuronal Dynamics 3.2. Nullclines of reced HH model d 0 stimulus F( u, ) RI( t) d u-nullcline G( u, ) 0 -nullcline Stable fixed point
38 Neuronal Dynamics 3.2. Phase Plane Analysis 2-dimensional equation stimulus F( u, ) RI( t) d G( u, ) Enables graphical analysis! Important role of Application to neuron models - nullclines - flo arros
39 Week 3 part 3: Analysis of a 2D neuron model 3.1 From Hodgkin-Huxley to 2D 3.2 Phase Plane Analysis - Role of nullcline 3.3 Analysis of a 2D Neuron Model - pulse input - constant input -MathDetour 3: Stability of fixed points 3.4 Type I and II Neuron Models (next eek)
40 Neuronal Dynamics 3.3. Analysis of a 2D neuron model 2-dimensional equation stimulus F( u, ) RI( t) 2 important input scenarios d G( u, ) - Pulse input Enables graphical analysis! - Constant input
41 Neuronal Dynamics D neuron model : Pulse input d u F ( u, ) RI d G ( u, ) pulse input
42 Neuronal Dynamics 3.3. FitzHugh-Nagumo Model : Pulse input 1 F( u, ) RI( t) u u 3 RI( t) 3 d 0 I(t)=0 d G ( u, ) b b u 0 1 u pulse input I(t) Pulse input: jump of voltage 0
43 Neuronal Dynamics 3.3. FitzHugh-Nagumo Model : Pulse input FN model ith b 0.9; b Pulse input: jump of voltage/initial condition Image: Neuronal Dynamics, Gerstner et al., Cambridge Univ. Press (2014)
44 Neuronal Dynamics 3.3. FitzHugh-Nagumo Model Pulse input: DONE! d 0 - jump of voltage - ne initial condition I(t)=0 - spike generation for large input pulses 2 important input scenarios constant input: u - graphics? - spikes? No - repetitive firing? 0
45 Neuronal Dynamics 3.3. FitzHugh-Nagumo Model: Constant input F( u, ) RI 0 d 0 -nullcline 1 u u 3 RI 3 0 d G ( u, ) b b u 0 1 Intersection point (fixed point) I(t)=I 0 u -moves -changes Stability 0 u-nullcline
46 NOW Exercise 2.1: Stability of Fixed Point in 2D u d 0 d u - calculate stability Exercises: - compare 2.1no! dx x I(t)=I 0 u 2.2 homeork Next lecture: 11:42 0
47 Week 3 part 3: Analysis of a 2D neuron model 3.1 From Hodgkin-Huxley to 2D 3.2 Phase Plane Analysis - Role of nullcline 3.3 Analysis of a 2D Neuron Model - pulse input - constant input -MathDetour 3: Stability of fixed points 3.4 Type I and II Neuron Models (next eek)
48 Neuronal Dynamics Detour 3.3 : Stability of fixed points. d b b u 0 1 d 0 -nullcline stable? I(t)=I 0 u F( u, ) RI 0 0 u-nullcline
49 Neuronal Dynamics 3.3 Detour. Stability of fixed points 2-dimensional equation stimulus F( u, ) RI 0 d G( u, ) Ho to determine stability of fixed point?
50 Neuronal Dynamics 3.3 Detour. Stability of fixed points stimulus d 0 au I 0 d cu I(t)=I 0 u 0 unstable saddle stable
51 Neuronal Dynamics 3.3 Detour. Stability of fixed points F( u, ) RI 0 d 0 -nullcline d G ( u, ) stable? zoom in: I(t)=I 0 u 0 unstable stable saddle Math derivation u-nullcline no
52 Neuronal Dynamics 3.3 Detour. Stability of fixed points F( u, ) RI 0 Fixed point at ( u, ) 0 0 At fixed point d G ( u, ) 0 F( u, ) RI G( u, ) 0 0 zoom in: x u u 0 y 0
53 Neuronal Dynamics 3.3 Detour. Stability of fixed points F( u, ) RI 0 Fixed point at ( u, ) 0 0 At fixed point d G ( u, ) 0 F( u, ) RI G( u, ) 0 0 zoom in: x u u 0 y 0 dx F x u F y dy G x G y u
54 Neuronal Dynamics 3.3 Detour. Stability of fixed points Linear matrix equation Search for solution To solution ith Eigenvalues, F u G F G F G u u
55 Neuronal Dynamics 3.3 Detour. Stability of fixed points Linear matrix equation Search for solution Stability requires: 0 and 0 To solution ith Eigenvalues, F u G F u and G 0 F G F G u u F G F G u u 0
56 Neuronal Dynamics 3.3 Detour. Stability of fixed points stimulus d 0 au I 0 d cu / I(t)=I 0 u 0 unstable saddle stable
57 Neuronal Dynamics 3.3 Detour. Stability of fixed points 2-dimensional equation F( u, ) RI 0 No Back: d G( u, ) Application to our neuron model Stability characterized by Eigenvalues of linearized equations
58 Neuronal Dynamics 3.3. FitzHugh-Nagumo Model: Constant input F( u, ) RI 0 d 0 -nullcline 1 u u 3 RI 3 0 d G ( u, ) b b u 0 1 Intersection point (fixed point) I(t)=I 0 u -moves -changes Stability 0 u-nullcline
59 Neuronal Dynamics 3.3. FitzHugh-Nagumo Model: Constant input F( u, ) RI 0 d 0 -nullcline 1 u u 3 RI 3 0 d G ( u, ) b b u 0 1 Intersection point (fixed point) I(t)=I 0 u -moves -changes Stability 0 u-nullcline
60 Neuronal Dynamics 3.3. FitzHugh-Nagumo Model : Constant input b 0.9; b 1.0; RI 2 FN model ith constant input: u-nullcline moves limit cycle f f-i curve Image: Neuronal Dynamics, Gerstner et al., CUP (2014) I0
61 Neuronal Dynamics Quiz 3.4. A. Short current pulses. In a 2-dimensional neuron model, the effect of a delta current pulse can be analyzed [ ] By moving the u-nullcline vertically upard [ ] By moving the -nullcline vertically upard [ ] As a potential change in the stability or number of the fixed point(s) [ ] As a ne initial condition [ ] By folloing the flo of arros in the appropriate phase plane diagram B. Constant current. In a 2-dimensional neuron model, the effect of a constant current can be analyzed [ ] By moving the u-nullcline vertically upard [ ] By moving the -nullcline vertically upard [ ] As a potential change in the stability or number of the fixed point(s) [ ] By folloing the flo of arros in the appropriate phase plane diagram
62 Computer exercise no Can e understand the dynamics of the 2D model? The END for today No: computer exercises Type I and type II models ramp input/ constant input f f-i curve f f-i curve I0 I0 I0
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