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Neurophysics Part 1: Neural encoding and decoding (Ch 1-4) Stimulus to response (1-2) Response to stimulus, information in spikes (3-4) Part 2: Neurons and Neural circuits (Ch 5-7) Classical neuron model (5) Extensions (6) Neural networks (7) Part 3: Adaptation and learning (Ch 8-10) Synaptic plasticity (8) Classical conditioning and RL (9) Pattern recognition and machine learning methods (10)

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Chapter 1

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Outline Neurons Firing rate Tuning curves Deviation from the mean: statistical description –Spike triggered average –Point process, Poisson process Poisson process –Homogeneous, Inhomogeneous –Experimental validation –shortcomings

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Properties of neurons Axon, dendrite Ion channels Membrane rest potential Action potential, refractory period

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Synapses, Ca influx, release of neurotransmitter, opening of post-synaptic channels

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Recording neuronal responses Intracellular recording –Sharp glass electrode or patch electrode –Typically in vitro Extracellular recording –Typically in vivo

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From stimulus to response Neurons respond to stimulus with train of spikes Response varies from trial to trial: –Arousal, attention –Randomness in the neuron and synapse –Other brain processes Population response Statistical description –Firing rate –Correlation function –Spike triggered average –Poisson model

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Spike trains and firing rates

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For Δ t ! 0, each interval contains 0,1 spike. Then, r(t) averaged over trials is the probability of any trial firing at time t. B: 100 ms bins

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C: Sliding rectangular window D: Sliding Gaussian window

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Causal window Temporal averaging with windows is non-causal. A causal alternative is w(t)=[ 2 t e - t ] + E: causal window

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Tuning curves For sensory neurons, the firing rate depends on the stimulus s Extra cellular recording V1 monkey Response depends on angle of moving light bar Average over trials is fitted with a Gaussian

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Motor tuning curves Extra cellular recording of monkey primary motor cortex M1 in arm-reaching task. Average firing rate is fitted with

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Retinal disparity Retinal disparity is location of object on retina, relative to the fixation point. Some neurons in V1 are sensitive to disparity.

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Spike-count variability Tuning curves model average behavior. Deviations of individual trials are given by a noise model. –Additive noise is independent of stimulus r=f(s)+ –Multiplicative noise is proportional to stimulus r=f(s) statistical description –Spike triggered average –Correlations

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Spike triggered average or reverse correlation What is the average stimulus that precedes a spike?

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Electric fish Left: electric signal and response of sensory neuron. Right: C( )

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Multi-spike triggered averages A: spike triggered average shows 15 ms latency; B: two- spike at 10 +/- 1 ms triggered average yields sum of two one-spike triggered averages; C: two-spike at 5 +/- 1 ms triggered average yields larger response indicating that multiple spikes may encode stimuli.

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Spike-train statistics If spikes are described as stochastic events, we call this a point process: P(t 1,t 2,…,t n )=p(t 1,t 2,…,t n )(Δ t) n The probability of a spike can in principle depend on the whole history: P(t n |t 1,…,t n-1 ) If the probability of a spike only depends on the time of the last spike, P(t n |t 1,…,t n-1 )=P(t n |t n-1 ) it is called a renewal process. If the probability of a spike is independent of the history, P(t n |t 1,…,t n-1 )=P(t n ), it is called a Poisson process.

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The Homogeneous Poisson Process The probability of n spikes in an interval T can be computed by dividing T in M intervals of size Δ t Right: rT=10, The distribution Approaches A Gaussian in n:

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Suppose a spike occurs at t I, what is the probability that the next spike occurs at t I+1 ? Mean inter-spike interval: Variance: Coefficient of variation: Inter-spike interval distribution

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Spike-train autocorrelation function Cat visual cortex. A: autocorrelation histograms in right (upper) and left (lower) hemispheres, show 40 Hz oscillations. B: Cross-correlation shows that these oscillations are synchronized. Peak at zero indicates synchrony at close to zero time delay

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Autocorrelation for Poisson process

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Inhomogeneous Poisson Process Divide the interval [t i,t i+1 ] in M segments of length Δ t. The probability of no spikes in [t i,t i+1 ] is

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The probability of spikes at times t 1,…t n is:

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Poisson spike generation Either –Choose small bins Δ t and generate with probability r(t)Δ t, or –Choose t i+1 -t I from p( )=r exp(-r ) Second method is much faster, but works for homogeneous Poisson processes only It is further discussed in an exercise.

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Model of orientation-selective neuron in V1 Top: orientation of light bar as a function of time. Middle: Orientation selectivity Bottom: 5 Poisson spike trials.

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Experimental validation of Poisson process: spike counts Mean spike count and variance of 94 cells (MT macaque) under different stimulus conditions. Fit of n 2 =A B yield A,B typically between 1-1.5, whereas Poisson yields A=B=1. variance higher than normal due to anesthesia.

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Experimental validation of Poisson process: ISIs Left: ISI of MT neuron, moving random dot image does not obey Poisson distribution 1.31 Right: Adding random refractory period (5 § 2 ms) to Poisson process restores similarity. One can also use a Gamma distribution

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Experimental validation of Poisson process: Coefficient of variation MT and V1 macaque.

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Shortcomings of Poisson model Poisson + refractory period accounts for much data but –Does not account difference in vitro and in vivo: neurons are not Poisson generators –Accuracy of timing (between trials) often higher than Poisson –Variance of ISI often higher than Poisson –Bursting behavior

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Types of coding: single neuron description Independent-spike code: all information is in the rate r(t). This is a Poisson process Correlation code: spike timing is history dependent. For instance a renewal process p(t i+1 |t i ) Deviation from Poisson process typically less than 10 %.

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Types of coding: neuron population Information may be coded in a population of neurons Independent firing is often valid assumption, but –Correlated firing is sometimes observed –For instance, Hippocampal place cells spike timing phase relative to common (7-12 Hz) rhythm correlates with location of the animal

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Types of coding: rate or temporal code? Stimuli that change rapidly tend to generate precisely timed spikes

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Chapter summary Neurons encode information in spike trains Spike rate –Time dependent r(t) –Spike count r –Trial average Tuning curve as a relation between stimulus and spike rate Spike triggered average Poisson model Statistical description: ISI histogram, C_V, Fano, Auto/Cross correlation Independent vs. correlated neural code

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Appendix A Power spectrum of white noise If Q_ss(t)=sigma^2 \delta(t) then Q_ss(w)=sigma^2/T Q_ss(w)=|s(w)|^2 36

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