Biological Modeling of Neural Networks: Week 15 – Population Dynamics: The Integral –Equation Approach Wulfram Gerstner EPFL, Lausanne, Switzerland 15.1.

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Biological Modeling of Neural Networks: Week 15 – Population Dynamics: The Integral –Equation Approach Wulfram Gerstner EPFL, Lausanne, Switzerland 15.1 Populations of Neurons - review: homogeneous population - review: parameters of single neurons 15.2 Integral equation - aim: population activity - renewal assumption 15.3 Populations with Adaptation - Quasi-renewal theory Coupled populations - self-coupling - coupling to other populations Week 15 – Integral Equation for population dynamics

Neuronal Dynamics – Brain dynamics is complex motor cortex frontal cortex to motor output neurons 3 km of wire 1mm Week 15-part 1: Review: The brain is complex

Homogeneous population: -each neuron receives input from k neurons in the population -each neuron receives the same (mean) external input (could come from other populations) -each neuron in the population has roughly the same parameters Week 15-part 1: Review: Homogeneous Population excitation inhibition Example: 2 populations

I(t) Week 15-part 1: Review: microscopic vs. macroscopic microscopic: -spikes are generated by individual neurons -spikes are transmitted from neuron to neuron macroscopic: -Population activity A n (t) -Populations interact via A n (t)

I(t) Week 15-part 1: Review: coupled populations

Populations: - pyramidal neurons in layer 5 of barrel D1 - pyramidal neurons in layer 2/3 of barrel D1 - inhibitory neurons in layer 2/3 of barrel D1 Week 15-part 1: Example: coupled populations in barrel cortex Neuronal Populations = a homogeneous group of neurons (more or less) neurons per group (Lefort et al., NEURON, 2009) different barrels and layers, different neuron types  different populations

-Extract parameters for SINGLE neuron (see NEURONAL DYNAMICS, Chs. 8-11; review now) -Dynamics in one (homogeneous) population Today, parts 15.2 and 15.3! -Dynamics in coupled populations -Understand Coding in populations -Understand Brain dynamics Sensory input Week 15-part 1: Global aim and strategy

i Spike emission potential threshold Input I(t) 3 arbitrary linear filters Week 15-part 1: Review: Spike Response Mode (SRM)

SRM + escape noise = Generalized Linear Model (GLM) i j -spikes are events -spike leads to reset/adapt. currents/increased threshold -strict threshold replaced by escape noise: spike exp Jolivet et al., J. Comput. NS, 2006

threshold current Input current: -Slow modulation -Fast signal Pozzorini et al. Nat. Neuros, 2013 Adaptation - extends over several seconds - power law

Adaptation current Dyn. threshold Mensi et al. 2012, Jolivet et al Week 15-part 1: Predict spike times + voltage with SRM/GLM

threshold current Dynamic threshold Dyn. Threshold unimportant Dyn. Threshold VERY IMPORTANT Fast Spiking Pyramidal N. Different neuron types have different parameters Dyn. Threshold important Non-fast Spiking Mensi et al Week 15-part 1: Extract parameter for different neuron types

-We can extract parameters for SINGLE neuron (see NEURONAL DYNAMICS, Chs. 8-11) -Different neuron types have different parameters  Model Dynamics in one (homogeneous) population Today, parts 15.2 and 15.3!  Couple different populations Today, parts 15.4! Sensory input Week 15-part 1: Summary and aims Eventually: Understand Coding and Brain dynamics

Biological Modeling of Neural Networks: Week 15 – Population Dynamics: The Integral –Equation Approach Wulfram Gerstner EPFL, Lausanne, Switzerland 15.1 Populations of Neurons - review: homogeneous population - review: parameters of single neurons 15.2 Integral equation - aim: population activity - renewal assumption 15.3 Populations with Adaptation - Quasi-renewal theory Coupled populations - self-coupling - coupling to other populations Week 15 – Integral Equation for population dynamics

I(t) Population of Neurons? Single neurons - model exists - parameters extracted from data - works well for step current - works well for time-dep. current Population equation for A(t) Week 15-part 2: Aim: from single neuron to population -existing models - integral equation

h(t) I(t) ? A(t) potential A(t) t population activity Question: – what is correct? Poll!!! A(t) Wilson-Cowan Benda-Herz LNP current Ostojic-Brunel Paninski, Pillow Week 15-part 2: population equation for A(t) ?

Experimental data : Tchumatchenko et al See also: Poliakov et al Week 15-part 2: A(t) in experiments: step current, (in vitro)

25000 identical model neurons (GLM/SRM with escape noise) parameters extracted from experimental data response to step current I(t) h(t) potential simulation data : Naud and Gerstner, 2012 Week 15-part 2: A(t) in simulations : step current

Week 15-part 2: A(t) in theory: an equation? Can we write down an equation for A(t) ? Simplification: Renewal assumption Potential (SRM/GLM) of one neuron h(t) Sum over all past spikes Keep only last spike  time-dependent renewal theory

Week 15-part 2: Renewal theory (time dependent) Potential (SRM/GLM) of one neuron last spike Keep only last spike  time-dependent renewal theory

Week 15-part 2: Renewal model –Example Example: nonlinear integrate-and-fire model deterministic part of input noisy part of input/intrinsic noise  escape rate Example: exponential stochastic intensity t

escape process A u(t) Survivor function t escape rate Interval distribution Survivor function escape rate u Week 15-part 2: Renewal model - Interspike Intervals

Interval distribution Survivor function escape rate Week 15-part 2: Renewal model – Integral equation population activity Image: Gerstner et al., Neuronal Dynamics, Cambridge Univ. Press (2014) Gerstner 1995, 2000; Gerstner&Kistler 2002 See also: Wilson&Cowan 1972 Blackboard!

Week 15-part 2: Integral equation – example: LIF input population of leaky integrate-and-fire neurons w. escape noise simulation Image: Gerstner et al., Neuronal Dynamics, Cambridge Univ. Press exact for large population N  infinity

Biological Modeling of Neural Networks: Week 15 – Population Dynamics: The Integral –Equation Approach Wulfram Gerstner EPFL, Lausanne, Switzerland 15.1 Populations of Neurons - review: homogeneous population - review: parameters of single neurons 15.2 Integral equation - aim: population activity - renewal assumption 15.3 Populations with Adaptation - Quasi-renewal theory Coupled populations - self-coupling - coupling to other populations Week 15 – Integral Equation for population dynamics

I(t) Population of adapting Neurons? Single neurons - model exists - parameters extracted from data - works well for step current - works well for time-dep. current Population equation for A(t) - existing modes - integral equation Week 15-part 3: Neurons with adapation Real neurons show adapation on multipe time scales

h(t) I(t) ? A(t) potential A(t) t population activity A(t) Wilson-Cowan Benda-Herz LNP current Week 15-part 3: population equation for A(t) ?

I(t) Rate adapt. optimal filter (rate adaptation) Benda-Herz Week 15-part 3: neurons with adaptation – A(t) ? simulation LNP theory

I(t) Population of Adapting Neurons? 1) Linear-Nonlinear-Poisson (LNP): fails! 2) Phenomenological rate adaptation: fails! 3) Time-Dependent-Renewal Theory ? Week 15-part 3: population equation for adapting neurons ?  Integral equation of previous section

Spike Response Model (SRM) i Spike emission potential Input I(t) h(t)

interspike interval distribution last firing time Gerstner 1995, 2000; Gerstner&Kistler 2002 See also: Wilson&Cowan 1972

Week 15-part 3: population equation for adapting neurons ?

I Rate adapt. Week 15-part 3: population equation for adapting neurons ? OK in late phaseOK in initial phase

I(t) Population of Adapting Neurons 1) Linear-Nonlinear-Poisson (LNP): fails! 2) Phenomenological rate adaptation: fails! 3) Time-Dependent-Renewal Theory: fails! 4) Quasi-Renewal Theory!!! Week 15-part 3: population equation for adapting neurons

and h(t) Expand the moment generating functional g(t-s) Week 15-part 3: Quasi-renewal theory Blackboard!

I Rate adapt. Quasi-Renewal RA OK in initial AND late phase

Week 15-part 3: Quasi-renewal theory Image: Gerstner et al., Neuronal Dynamics, Cambridge Univ. Press PSTH or population activity A(t) predicted by quasi-renewal theory

I(t) Population of Adapting Neurons? 1) Linear-Nonlinear-Poisson (LNP): fails! 2) Phenomenological rate adaptation: fails! 3) Time-Dependent-Renewal Theory : fails! 4) Quasi-Renewal Theory: works!!!! Week 15-part 3: population equation for adapting neurons

Biological Modeling of Neural Networks: Week 15 – Population Dynamics: The Integral –Equation Approach Wulfram Gerstner EPFL, Lausanne, Switzerland 15.1 Populations of Neurons - review: homogeneous population - review: parameters of single neurons 15.2 Integral equation - aim: population activity - renewal assumption 15.3 Populations with Adaptation - Quasi-renewal theory Coupled populations - self-coupling - coupling to other populations Week 15 – Integral Equation for population dynamics

Week 15-part 4: population with self-coupling Same theory works - Include the self-coupling in h(t) Potential (SRM/GLM) of one neuron h(t) J0J0

Week 15-part 4: population with self-coupling Image: Gerstner et al., Neuronal Dynamics, Cambridge Univ. Press

Week 15-part 4: population with self-coupling Image: Gerstner et al., Neuronal Dynamics, Cambridge Univ. Press J0J0

Week 15-part 4: population with self-coupling instabilities and oscillations noise level delay stable asynchronous state Image: Gerstner et al., Neuronal Dynamics, Cambridge Univ. Press

Week 15-part 4: multipe coupled populations Same theory works - Include the cross-coupling in h(t) Potential (SRM/GLM) of one neuron in population n h n (t) J 11 J 22 J 21 J 12

-We can extract parameters for SINGLE neuron (see NEURONAL DYNAMICS, Chs. 8-11) -Different neuron types have different parameters  Model Dynamics in one (homogeneous) population Today, parts 15.2 and 15.3!  Couple different populations Today, parts 15.4! Sensory input Week 15-part 1: Summary and aims Eventually: Understand Coding and Brain dynamics

The end Reading: Chapter 14 of NEURONAL DYNAMICS, Gerstner et al., Cambridge Univ. Press (2014)