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Neural Coding and Integrate-And-Fire Model motor Introduction to computationalneuroscience. cortex Doctoral school Oct-Nov 2006 association visual cortex cortex to motor Wulfram Gerstner output http://diwww.epfl.ch/w3mantra/ Computational Neuroscience 10 000 neurons Signal: 3 km wires action potential (spike) 1mm behavior behavior action potential neurons signals computational model molecules ion channels Swiss Federal Institute of Technology Lausanne, EPFL Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Hodgkin-Huxley type models Integrate-and-fire type models Spike emission j i u i J Spike reception -70mV action potential Na+ K+ -spikes are events Ca2+ -threshold Ions/proteins -spike/reset/refractoriness 1 Models of synaptic Plasticity Computational Neuroscience pre post i Synapse j behavior Neurons synapses computational model molecules ion channels Ecole Polytechnique Fédérale de Lausanne, EPFL Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Introduction to computationalneuroscience Background: What is brain-style computation? Lecture 1: Passive membrane and Integrate-and-Fire model Brain Computer Lecture 2: Hodgkin-Huxley models (detailed models) Lecture 3: Two-dimensional models (FitzHugh Nagumo) Lecture 4: synaptic plasticity Lecture 5: noise, network dynamics, associative memory Wulfram Gerstner http://diwww.epfl.ch/w3mantra/ Systems for computing and information processing Systems for computing and information processing Brain Computer Brain Computer memory Tasks: CPU slow Mathematical fast æ7p ö input 5cosç ÷ è 5 ø Distributed architecture Von Neumann architecture 10 fast Real world (10 proc. Elements/neurons) 1 CPU slow (10 10 transistors) E.g. complex scenes No separation of processing and memory 2 Systems for computing and information processing Introduction to computationalneuroscience Brain Computer Lecture 1: Passive membrane and Integrate-and-Fire model Lecture 2: Hodgkin-Huxley models (detailed models) Clear separation: Where is the program? software (pogram)/hardware Lecture 3: Two-dimensional models (FitzHugh Nagumo) Lecture 4: synaptic plasticity Clear separation: Lecture 5: noise, network dynamics, associative memory Where is the memory? memory/processing In the synapticconnections Wulfram Gerstner http://diwww.epfl.ch/w3mantra/ Why spiking neuron models? The Problem of Neuronal Coding Spikes versus rates nsp T stim The Problem of Neuronal Coding n ( t ; t + T ) Rate n = sp T Book: Spiking Neuron Models Rate defined as temporal average Swiss Federal InstituteChapter of 1.4 Technology Lausanne, EPFL Swiss Federal Institute of Technology Lausanne, EPFL Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne The Problem of Neuronal Coding The problem of neural coding: Rate defined as average over stimulus repetitions population activity - rate defined by population average t Peri-Stimulus Time Histogram Stim(t) PSTH(t) t ? I(t) K=500 trials population dynamics? t n (t ; t + D t ) PSTH ( t ) = n(t;t + Dt) K D t population A(t) = activity NDt Swiss Federal Institute of Technology Lausanne, EPFL Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne 3 Relevance of temporal aspects-1 The problem of neural coding: temporal codes Reverse Correlations Time to first spike after input I(t) t fluctuating input correlations Phase with respect to oscillation Swiss Federal Institute of Technology Lausanne, EPFL Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Reverse-Correlation Experiments (simulations) Relevance of temporal aspects-2 Stimulus Reconstruction DI (t) I(t) after 100025000 spikes spikes fluctuating input Swiss Federal Institute of Technology Lausanne, EPFL Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Relevance of temporal aspects-3 Stimulus Reconstruction Intrinsic reliability of neurons model vs exper. 3 repetitions I(t) m s Bialek et al C Exp. Data: See also: Mainen&Sejnowski A. Rauch, H.Lüscher, U. Berne 4 Introduction to computationalneuroscience If we want to avoid prior assumptions about neural coding, Lecture 1: we need to model neurons on the level of action potentials: Passive membrane and Integrate -and-Fire model -The problem of neural coding spiking neuron models -The passive membrae -Leaky ingreate-and-fire model -Generalized integrate-and-fire model -Quality of integrate-and-fire models -Coding revisited Wulfram Gerstner http://diwww.epfl.ch/w3mantra/ Integrate-and-fire type models Introduction to computationalneuroscience Spike emission Lecture 1: j i Passive membrane and Integrate -and-Fire model ui -The problem of neural coding J -The passive membrae Spike reception -Leaky ingreate-and-fire model -Generalized integrate-and-fire model -Quality of integrate-and-fire models -Coding revisited -spikes are events Subthreshold regime -threshold - linear -spike/reset/refractoriness - passive membrane Wulfram Gerstner http://diwww.epfl.ch/w3mantra/ Subthreshold regime – passive membrane Passive Membrane Model j i j i ui ui Spike reception: EPSP Spike reception I Subthreshold regime - linear - passive membrane 5 Passive Membrane Model Passive Membrane Model I(t) Time-dependent input j i i ui u I(t) I u d t × u = -(u - u ) + RI (t) Blackboard : dt rest Blackboard : Derive equation d Voltage scale t × V = -V + RI(t); V = (u - u ) dt rest Passive Membrane Passive Membrane Model-now exercises j i i u u i Free solution: exponential decay I(t) i Step current input: u I rest d Start at t 0 with u1 d t × u = -(u -u ); t × u = -(u - u ) + RI (t) linear dt rest dt rest Blackboard : t -t0 u(t) = urest + (u0 -urest )exp(- ) free solution t Passive Membrane Model-now exercises Chapter 4: Formal Spiking models j i Integrate-and-Fire model ui impulse reception: impulse response I function BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler d Cambridge University Press, 2002 t × u = -(u - urest ) + RI (t) linear dt Chapter 4 6 Integrate-and-fire type models Integrate-and-fire Model Spike emission Spike emission j i j i ui ui Spike reception: EPSP J J reset Spike reception I J d linear -spikes are events t × u = -(u - urest ) + RI (t) -threshold dt -spike/reset/refractoriness ui (t) = J Þ Fire+reset threshold Integrate-and-fire type models Integrate-and-fire Models I(t) Time-dependent input d LIF t × u = -(u - urest ) + RI (t) dt If firing: u ® ureset i u I=0 I>0 d d J u u dt dt I(t) Blackboard J u -spikes are events J repetitive J u -threshold u resting -spike/reset/refractoriness J t t Integrate-and-fire Models Nonlinear Integrate-and-fire Model d I=0 I>0 t × u = -(u - urest ) + RI (t) LIF d d dt u u d dt dt t × u = F (u) + RI(t) NLIF dt If firing: u ® u reset u u q q d t × u = F (u) + RI (t) NONlinear dt u(t)= q Þ Fire+reset threshold 7 Nonlinear Integrate-and-fire Model Nonlinear Integrate-and-fire Model I=0 I>0 j Spike emission d d i u u u dt dt i J F reset I J u u J J d d Quadratic I&F: t × u = F (u) + RI (t) NONlinear t × u = F (u) + RI (t) NONlinear F(u) = c (u - c ) 2 + c dt dt 2 1 0 ui (t) = J Þ Fire+reset threshold u(t) = J Þ Fire+reset threshold Nonlinear Integrate-and-fire Model Nonlinear Integrate-and-fire Model I=0 I>0 I=0 I>0 d d d d u u u u dt dt dt dt u u u u J J J J exponentialI&F: d Quadratic I&F: d F(u) = -(u - u ) t × u = F (u) + RI (t) F(u) = c (u - c ) 2 + c t × u = -F((uu-) +urestRI)(+t) RINONlinear(t) rest dt 2 1 0 dt + c exp(u -J) exponentialI&F: 0 u(t) = J Þ Fire+reset u(t) = J Þ Fire+reset threshold F(u) = -(u - u rest ) + c 0 exp(u -J) Nonlinear Integrate-and-fire Model Where is the firing threshold? Integrate-and Fire type models I=0 I>0 d d Where is the firing threshold? u u dt dt Leaky integrate-and-fire (LIF) Strict voltage threshold - by construction - spike threshold = reset condition u u resting J q q Nonlinear integrate-and-fire (eIF) u There is no strict firing threshold J J - firing depends on input - exact reset condition of minor relevance t t d t × u = F(u) + RI(t) dt 8 Nonlinear Integrate-and-fire Model Leaky Integrate-and-fire Model Where is the firing threshold? I=0 I>0 d LIF d d t × u = -(u - u ) + RI u u rest 0 If firing: u ® ureset dt dt dt repetitive u J u u t q u Exercises 2+3 Stable fixed point What is the firing rate? now! t f=g(I) Exercise 2+3 now! d t × u = F(u) + RI(t) dt Integrate-and-fire type models Integrate-and Fire type models Leaky integrate-and-fire (LIF) Spike emission Strict voltage threshold j i u i J Spike reception Nonlinear Spike Response integrate-and-fire Model (SRM) no strict firing threshold Includes refractoriness -spikes are events Subthreshold regime -threshold - linear -spike/reset/refractoriness - passive membrane Spike Response Model and refractoriness Spike emission Integrate&Fire: Where is the firing threshold? J ( t - t ' ) 1: Introduction ^ 2: Integrate-and-Fire (and generalisations) threshold ti u(t)= J 3. Quality of Integrate-and-Fire models du(t) Firing: t'= t ³ 0 dt Wulfram Gerstner http://diwww.epfl.ch/w3mantra/ 9 Validation of neuron models Real neuron vs. Nonlinear I&F Cortical layer V pyramidal neuron Spike J I(t) threshold model exponential Integrate-and-fire Model Measurement of the dynamic IV curve I-w=0 d d I-w>0 u u dt dt firing Not firing u u Q Qq voltage d u-J t × u = u -urest + Dexp( D ) + R[I(t)] Experiments: Theory: dt Sandrine Lefort Laurent Badel Carl Petersen Magnus Richardson u(t)= Q Þ Fire --- reset of u Experimental method Comparison of the voltage traces filtered current 3ms and 10ms I [pA] Spikes predicted with 5ms accuracy V [mV] model 77% of spikes Layer 5 pyramidal somatosensory cortex 10 The after-spike neuronal response function The post -spike neuronal response function EIF with a dynamic threshold, resting potential and time constant Comparison of the model and experiment for the voltage trace 82% of spikes dV ( EVL - ) D-æöVV =+TTexp ç÷ dt ttLLTèøD dV (VVTT¥ - ) T = Fuortes and Mantegazzini (1962) dt t VT dE ( EE- ) Jolivet et al (2003) L = LL¥ Rauch et al (2003) With a high reset: dt t Brette and Gerstner (2005) EL VVreT=-39mV>¥ =-41mV Theory: Laurent Badel Magnus Richardson tt- dtL ( LL¥ ) = Wehmeier et al (1989) dt t tL Validation of neuron models Integrate&Fire: (Clopath et al.
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