of Neural Networks:
Biological Modeling
of Neural Networks:
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Wulfram Gerstner
EPFL, Lausanne, Switzerland
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Neuronal Dynamics –
9
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Adaptation
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Response to
Step currents,
Exper. Data,
Markram et al.
(2004)
I(t)
Biological Modeling
of Neural Networks:
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Wulfram Gerstner
EPFL, Lausanne, Switzerland
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Firing patterns and adaptation
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:
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Neuronal Dynamics –
9
.2
Adaptive Exponential I&F
Exponential I&F
+
1
a
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a
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v
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= AdEx
Blackboard !
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Response to
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Exper. Data,
Markram et al.
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Response to
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A
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Image:
Neuronal Dynamics,
Gerstner et al.
Cambridge (2002)
A
d
E
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Can we understand the different firing patterns?
P
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a
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p
l
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y
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!
Neuronal Dynamics –
9
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Adaptive Exponential I&F
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[ ] constant
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Neuronal Dynamics –
Quiz 9.1. Nullclines of AdEx
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[ ] linear, slope 1
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1 minute
Restart at 9:38
Biological Modeling
of Neural Networks:
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Wulfram Gerstner
EPFL, Lausanne, Switzerland
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Firing patterns arise from different parameters!
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Neuronal Dynamics –
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.2 AdEx model and firing patterns
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Neuronal Dynamics –
Review
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Nonlinear Integrate-and-fire
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Neuronal Dynamics –
9
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AdEx with dynamic threshold
Neuronal Dynamics –
9
.2 Generalized
Integrate-and-fire
Biological Modeling
of Neural Networks:
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Wulfram Gerstner
EPFL, Lausanne, Switzerland
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Neuronal Dynamics –
9
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Adaptive leaky integrate-and-fire
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Neuronal Dynamics –
9
.3
Adaptive leaky I&F and SRM
Adaptive
leaky I&F
threshold
Gerstner et al.,
1993, 1996
u(t)
Neuronal Dynamics –
9
.3
Spike Response Model
(SRM)
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Neuronal Dynamics –
9
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Bursting in the SRM
E
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Integrate the above system of two differential
equations so as to rewrite the equations as
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Next lecture
at 9:57/10:15
threshold
Input
I(t)
S(t)
firing if
Gerstner et al.,
1993, 1996
Neuronal Dynamics –
9
.3
Spike Response Model
(SRM)
Neuronal Dynamics –
9.3
Spike Response Model
(SRM)
Linear filters for
- input
- threshold
- refractoriness
This lesson plan for Grade 8 focuses on graphing linear equations in slope-intercept form. It covers learners' characteristics, entry skills, state objectives, teacher-student interaction, substitution activities, and motivational exercises to enhance understanding. Utilizing visual aids and engaging activities, students will develop proficiency in graphing linear equations efficiently.
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Week 9 adaptation and firing patterns 9.1 Firing patterns and adaptation 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation Biological Modeling of Neural Networks: Week 9 Adaptation and firing patterns Wulfram Gerstner EPFL, Lausanne, Switzerland
Neuronal Dynamics 9.1Adaptation Step current input neurons show adaptation I(t) Data: Markram et al. (2004) 1-dimensional (nonlinear) integrate-and-fire model cannot do this!
Firing patterns: Response to Step currents, Exper. Data, Markram et al. (2004) I(t)
Week 9 adaptation and firing patterns 9.1 Firing patterns and adaptation 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation Biological Modeling of Neural Networks: Week 9 Adaptation and firing patterns Wulfram Gerstner EPFL, Lausanne, Switzerland
Neuronal Dynamics 9.2Adaptive Exponential I&F Add adaptation variables: Blackboard ! du dt u = u u + + ( ) exp( ) ( ) R w RI t Exponential I&F + 1 adaptation var. = AdEx rest k k dw dt = + f ( ) ( ) k a u u w b t t k k rest k k k f after each spike w SPIKE AND RESET AdEx model, Brette&Gerstner (2005): k jumps by an amount k b = = If u then resetto u u reset r
Firing patterns: Response to Step currents, Exper. Data, Markram et al. (2004) I(t)
Firing patterns: Response to Step currents, AdEx Model, Naud&Gerstner I(t) Image: Neuronal Dynamics, Gerstner et al. Cambridge (2002)
Neuronal Dynamics 9.2Adaptive Exponential I&F du dt dw dt u = u u + + ( ) exp( ) ( ) Rw RI t rest AdEx model = w b + f ( ) ( ) a u u t t w rest w f Phase plane analysis! Can we understand the different firing patterns?
Neuronal Dynamics Quiz 9.1. Nullclinesof AdEx du dt dw dt u = + + ( ) exp( ) ( ) u u Rw RI t rest = ( ) a u u w w rest A - What is the qualitative shape of the w-nullcline? [ ] constant [ ] linear, slope a [ ] linear, slope 1 [ ] linear + quadratic [ ] linear + exponential B - What is the qualitative shape of the u-nullcline? [ ] linear, slope 1 [ ] linear, slope 1/R [ ] linear + quadratic [ ] linear w. slope 1/R+ exponential 1 minute Restart at 9:38
Week 9 part 2b : Firing Patterns 9.1What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? Biological Modeling of Neural Networks: Week 9 Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland
after each spike AdEx model u is reset to ur du dt u = + + ( ) exp( ) ( ) u u Rw RI t rest dw dt = + f ( ) ( ) a u u w b t t w rest w f after each spike w jumps by an amount b parameter a slope of w-nullcline Can we understand the different firing patterns?
AdEx model phase plane analysis: large b du dt dw dt u = + + + ( ) exp( ) ( ) u u w RI t rest = + f ( ) ( ) a u u a=0 w b t t w rest w f u-nullcline b u is reset to ur
AdEx model phase plane analysis: small b du u u u dt dw a u u w b dt = + + + ( ) exp( ) ( ) w RI t rest = + f ( ) ( ) t t w rest w f adaptation u-nullcline b u is reset to ur
Quiz 9.2: AdEx model phase plane analysis du dt u = + + + ( ) exp( ) ( ) u u w RI t rest dw dt = + f ( ) ( ) a u u b t t w rest w f What firing pattern do you expect? (i) Adapting (ii) Bursting (iii) Initial burst (iv)Non-adapting u-nullcline b u is reset to ur
AdEx model phase plane analysis: a>0 du dt dw dt u = + + + ( ) exp( ) ( ) u u w RI t rest = + f ( ) ( ) a u u w b t t w rest w f u-nullcline b u is reset to ur
Neuronal Dynamics 9.2 AdExmodel and firing patterns after each spike u is reset to ur du dt u = + + ( ) exp( ) ( ) u u Rw RI t rest dw dt = + f ( ) ( ) a u u w b t t w rest w f after each spike w jumps by an amount b parameter a slope of w nullcline Firing patterns arise from different parameters! See Naud et al. (2008), see also Izikhevich (2003)
Neuronal Dynamics Review: Nonlinear Integrate-and-fire du dt = + ( ) f u ( ) RI t (1) = = If u then resetto u u (2) reset r Best choice of f : linear + exponential du dt u = + ( ) exp( ) u u rest BUT: Limitations need to add -Adaptation on slower time scales -Possibility for a diversity of firing patterns -Increased threshold after each spike -Noise
Neuronal Dynamics 9.2AdExwith dynamic threshold Add dynamic threshold: du dt u = + + ( ) exp( ) ( ) u u R w RI t rest k k Threshold increases after each spike = + f 1( ) t t 0 f
Neuronal Dynamics 9.2 GeneralizedIntegrate-and-fire du dt = + ( ) f u ( ) RI t = = If u then resetto u u reset r add -Adaptation variables -Possibility for firing patterns -Dynamic threshold -Noise
Week 9 part 3: Spike Response Model (SRM) 9.1What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? Biological Modeling of Neural Networks: Week 9 Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland
Exponential versus Leaky Integrate-and-Fire du dt u = + + ( ) exp( ) ( ) u u RI t Badel et al (2008) rest = 2mV du dt = u u + ( ) ( ) Leaky Integrate-and-Fire: Replace nonlinear kink by threshold RI t rest Reset if u=
Neuronal Dynamics 9.3Adaptive leaky integrate-and-fire du dt = + ( ) ( ) u u R w RI t rest k k dw dt = + f ( ) ( ) k a u u w b t t k k rest k k k f after each spike w jumps by an amount ( ) If u t then resetto u = SPIKE AND RESET k b k = u r Dynamic threshold
Neuronal Dynamics 9.3Adaptive leaky I&F and SRM du dt = u u + ( ) ( ) R w RI t Adaptive leaky I&F rest k k dw dt = + k k b t t f ( ) ( ) k a u u w k k rest k f Linear equation can be integrated! = t t + ( ) ( s I t f ( ) ( ) ) u t ds s f Spike Response Model (SRM) Gerstner et al. (1996) 0 = + t t f ( ) t ( ) 0 1 f
Neuronal Dynamics 9.3Spike Response Model (SRM) Gerstner et al., 1993, 1996 ( ) t i Input I(t) iu ( ) s Spike emission u(t) Arbitrary Linear filters potential ( )= t ( ) ( ) s I t + + u ( ) s ds u ' t t rest 0 ' t = + t t threshold ( ) t ( ') 0 1 ' t
Neuronal Dynamics 9.3Bursting in the SRM SRM with appropriate leads to bursting = t t + ( ) ( s I t + f ( ) ( ) ) u t ds s u rest f 0 = ( ) ( s S t + ( ) ( s I t + ( ) ) ) u t ds s ds s u rest 0 0
Exercise 1: from adaptive IF to SRM du dt = + ( ) ( ) u u w RI t rest = = If u then resetto u u r dw dt = + f ( ) w b t t w w f Next lecture at 9:57/10:15 Integrate the above system of two differential equations so as to rewrite the equations as potential 0 A what is ? B what is ? x s (iii) ( ) s I t ( )= t ( ) s S t + + ( ) s ds u ( ) s ds u rest 0 R s R s ( ) s ( ) s ( ) x s ( ) x s = = exp( ) exp( ) (i) (ii) s w w s ( ) (iv) Combi of (i) + (iii) = ) exp( [exp( )] C w
Neuronal Dynamics 9.3Spike Response Model (SRM) Gerstner et al., 1993, 1996 1( ) s Input I(t) ih S(t) ( ) s + = u ( ) s ( )= t ( ) ( ) s I t ') t ( + + u potential ( ) s ds u + ' t t rest 0 ' t = ( ) t t threshold 0 1 ' t = ( ) ( ) t u t firing if
Neuronal Dynamics 9.3Spike Response Model (SRM) + potential ( )= t u ( ) ( ) s I t + + ( ) ' s ds u t t rest 0 ' t Linear filters for - input - threshold - refractoriness threshold = + t t ( ) t ( ') 0 1 ' t
