Theoretical Neuroscience Physics 405, Copenhagen University Block 4, Spring 2007 John Hertz (Nordita) Office: rm Kc10, NBI Blegdamsvej Tel 3532 5236 (office)

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Presentation transcript:

Theoretical Neuroscience Physics 405, Copenhagen University Block 4, Spring 2007 John Hertz (Nordita) Office: rm Kc10, NBI Blegdamsvej Tel (office) (mobil) Texts: P Dayan and L F Abbott, Theoretical Neuroscience (MIT Press) W Gerstner and W Kistler, Spiking Neuron Models (Cambridge U Press)

Outline Introduction: biological background, spike trains Biophysics of neurons: ion channels, spike generation Synapses: kinetics, medium- and long-term synaptic modification Mathematical analysis using simplified models Network models: –noisy cortical networks –primary visual cortex –associative memory –oscillations in olfactory circuits

Lecture I: Introduction ca neurons/human brain 10 4 /mm 3 soma  m axon length ~ 4 cm total axon length/mm 3 ~ 400 m

Cell membrane, ion channels, action potentials Membrane potential: rest at ca -70 mv Na-K pump maintains excess K inside, Na outside Na in: V rises, more channels open  “spike”

Communication: synapses Integrating synaptic input:

Brain anatomy: functional regions

Visual system General anatomy Retina

Neural coding: firing rates depend on stimulus Visual cortical neuron: variation with orientation of stimulus

Neural coding: firing rates depend on stimulus Visual cortical neuron: variation with orientation of stimulus Motor cortical neuron: variation with direction of movement

Neuronal firing is noisy Motion-sensitive neuron in visual area MT: spike trains evoked by multiple presentations of moving random-dot patterns

Neuronal firing is noisy Motion-sensitive neuron in visual area MT: spike trains evoked by multiple presentations of moving random-dot patterns Intracellular recordings of membrane potential: Isolated neurons fire regularly; neurons in vivo do not:

Quantifying the response of sensory neurons spike-triggered average stimulus (“reverse correlation”)

Examples of reverse correlation Electric sensory neuron in electric fish: s(t) = electric field Motion-sensitive neuron in blowfly Visual system: s(t) = velocity of moving pattern in visual field Note: non-additive effect for spikes very close in time (  t < 5 ms 

Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval

Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t)

Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t)

Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t) Probability of firing for the first time in [t, t +  t)/  t :

Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t) Probability of firing for the first time in [t, t +  t)/  t :

Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t) Probability of firing for the first time in [t, t +  t)/  t : (interspike interval distribution)

Homogeneous Poisson process (2) Probability of exactly 1 spike in [0,T):

Homogeneous Poisson process (2) Probability of exactly 1 spike in [0,T): Probability of exactly 2 spikes in [0,T):

Homogeneous Poisson process (2) Probability of exactly 1 spike in [0,T): Probability of exactly 2 spikes in [0,T): … Probability of exactly n spikes in [0,T):

Homogeneous Poisson process (2) Probability of exactly 1 spike in [0,T): Probability of exactly 2 spikes in [0,T): … Probability of exactly n spikes in [0,T): Poisson distribution

Pprobability of n spikes in interval of duration T :

Poisson distribution Pprobability of n spikes in interval of duration T : Mean count:

Poisson distribution Pprobability of n spikes in interval of duration T : Mean count: variance:

Poisson distribution Pprobability of n spikes in interval of duration T : Mean count: variance: i.e., spikes

Poisson distribution Pprobability of n spikes in interval of duration T : Mean count: variance: i.e., spikes large :  Gaussian

Poisson distribution Pprobability of n spikes in interval of duration T : Mean count: variance: i.e., spikes large :  Gaussian

Poisson process (2): interspike interval distribution Exponential distribution: (like radioactive Decay)

Poisson process (2): interspike interval distribution Exponential distribution: (like radioactive Decay) Mean ISI:

Poisson process (2): interspike interval distribution Exponential distribution: (like radioactive Decay) Mean ISI: variance:

Poisson process (2): interspike interval distribution Exponential distribution: (like radioactive Decay) Mean ISI: variance: Coefficient of variation:

Poisson process (3): correlation function Spike train:

Poisson process (3): correlation function Spike train: mean:

Poisson process (3): correlation function Spike train: mean: Correlation function:

Stationary renewal process Defined by ISI distribution P(t)

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) :

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define Laplace transform:

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define Laplace transform: Solve:

Fano factor spike count variance / mean spike count for stationary Poisson process

Fano factor spike count variance / mean spike count for stationary Poisson process

Fano factor spike count variance / mean spike count for stationary Poisson process 

Fano factor spike count variance / mean spike count for stationary Poisson process  for stationary renewal process (prove this)

Nonstationary point processes

Nonstationary Poisson process: time-dependent rate r(t) Still have Poisson count distribution, F=1

Nonstationary point processes Nonstationary Poisson process: time-dependent rate r(t) Still have Poisson count distribution, F=1 Nonstationary renewal process: time-dependent ISI distribution = ISI probability starting at t 0

Experimental results (1) Correlation functions

Experimental results (1) Correlation functionsCount variance vs mean

Experimental results (2) ISI distribution

Experimental results (2) ISI distribution CV’s for many neurons

homework Prove that the ISI distribution is exponential for a stationary Poisson process. Prove that the CV is 1 for a stationary Poisson process. Show that the Poisson distribution approaches a Gaussian one for large mean spike count. Prove that F = CV 2 for a stationary renewal process. Show why the spike count distribution for an inhomogeneous Poisson process is the same as that for a homogeneous Poisson process with the same mean spike count.