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Theoretical Neuroscience. Physics 405, Copenhagen University Block 4, Spring 2007 John Hertz (Nordita) Office: rm Kc10, NBI Blegdamsvej Tel 3532 5236 (office) 2720 4184 (mobil) hertz@nordita.dk www.nordita.dk/~hertz/course.html Texts:
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Theoretical Neuroscience Physics 405, Copenhagen University Block 4, Spring 2007 John Hertz (Nordita) Office: rm Kc10, NBI Blegdamsvej Tel 3532 5236 (office) 2720 4184 (mobil) hertz@nordita.dk www.nordita.dk/~hertz/course.html Texts: P Dayan and L F Abbott, Theoretical Neuroscience (MIT Press) W Gerstner and W Kistler, Spiking Neuron Models (Cambridge U Press)http://diwww.epfl.ch/~gerstner/SPNM/SPNM.html
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 1011 neurons/human brain 104/mm3 soma 10-50 mm axon length ~ 4 cm total axon length/mm3 ~ 400 m
Cell membrane, ion channels, action potentials Na in: V rises, more channels open “spike” Membrane potential: rest at ca -70 mv Na-K pump maintains excess K inside, Na outside
Communication: synapses Integrating synaptic input:
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 Motion-sensitive neuron in blowfly Visual system: s(t) = velocity of moving pattern in visual field Electric sensory neuron in electric fish: s(t) = electric field Note: non-additive effect for spikes very close in time (Dt < 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 + Dt)/ Dt :
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 + Dt)/ Dt :
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 + Dt)/ Dt : (interspike interval distribution)
Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T):
Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T): Probabilityof exactly 2 spikes in [0,T):
Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T): Probabilityof exactly 2 spikes in [0,T): … Probabilityof exactly n spikes in [0,T):
Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T): Probabilityof exactly 2 spikes in [0,T): … Probabilityof exactly n spikes in [0,T): Poisson distribution
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 (like radioactive Decay) Exponential distribution:
Poisson process (2): interspike interval distribution (like radioactive Decay) Exponential distribution: Mean ISI:
Poisson process (2): interspike interval distribution (like radioactive Decay) Exponential distribution: Mean ISI: variance:
Poisson process (2): interspike interval distribution (like radioactive Decay) Exponential distribution: 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 t0
Experimental results (1) Correlation functions