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Neurophysics

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Title: Neurophysics


1
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)

2
Chapter 1
3
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

4
Properties of neurons
  • Axon, dendrite
  • Ion channels
  • Membrane rest potential
  • Action potential, refractory period

5
Synapses, Ca influx, release of neurotransmitter,
opening of post-synaptic channels
6
Recording neuronal responses
  • Intracellular recording
  • Sharp glass electrode or patch electrode
  • Typically in vitro
  • Extracellular recording
  • Typically in vivo

7
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

8
Spike trains and firing rates
9
For ? 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
10
C Sliding rectangular window D Sliding Gaussian
window
11
Causal window
  • Temporal averaging with windows is non-causal. A
    causal alternative is w(t)a2 t e-a t

E causal window
12
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

13
Motor tuning curves
  • Extra cellular recording of monkey primary motor
    cortex M1 in arm-reaching task. Average firing
    rate is fitted with

14
Retinal disparity
  • Retinal disparity is location of object on
    retina, relative to the fixation point.
  • Some neurons in V1 are sensitive to disparity.

15
Spike-count variability
  • Tuning curves model average behavior.
  • Deviations of individual trials are given by a
    noise model.
  • Additive noise is independent of stimulus
    rf(s)?
  • Multiplicative noise is proportional to stimulus
    rf(s) ?
  • statistical description
  • Spike triggered average
  • Correlations

16
Spike triggered average or reverse correlation
  • What is the average stimulus that precedes a
    spike?

17
Electric fish
  • Left electric signal and response of sensory
    neuron.
  • Right C(t)

18
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.

19
Spike-train statistics
  • If spikes are described as stochastic events, we
    call this a point process P(t1,t2,,tn)p(t1,t2,
    ,tn)(? t)n
  • The probability of a spike can in principle
    depend on the whole history P(tnt1,,tn-1)
  • If the probability of a spike only depends on the
    time of the last spike, P(tnt1,,tn-1)P(tntn-1)
    it is called a renewal process.
  • If the probability of a spike is independent of
    the history, P(tnt1,,tn-1)P(tn), it is called
    a Poisson process.

20
The Homogeneous Poisson Process
  • The probability of n spikes in an interval T can
    be computed by dividing T in M intervals of size
    ? t

Right rT10, The distribution Approaches A
Gaussian in n
21
Inter-spike interval distribution
  • Suppose a spike occurs at tI, what is the
    probability that the next spike occurs at tI1?
  • Mean inter-spike interval
  • Variance
  • Coefficient of variation

22
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
23
Autocorrelation for Poisson process
24
Inhomogeneous Poisson Process
  • Divide the interval ti,ti1 in M segments of
    length ? t.
  • The probability of no spikes in ti,ti1 is

25
  • The probability of spikes at times t1,tn is

26
Poisson spike generation
  • Either
  • Choose small bins ? t and generate with
    probability r(t)? t, or
  • Choose ti1-tI from p(t)r exp(-r t)
  • Second method is much faster, but works for
    homogeneous Poisson processes only
  • It is further discussed in an exercise.

27
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.

28
Experimental validation of Poisson process spike
counts
  • Mean spike count and variance of 94 cells (MT
    macaque) under different stimulus conditions.
  • Fit of sn2A ltngtB yield A,B typically between
    1-1.5, whereas Poisson yields AB1.
  • variance higher than normal due to anesthesia.

29
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

30
Experimental validation of Poisson process
Coefficient of variation
  • MT and V1 macaque.

31
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

32
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(ti1ti)
  • Deviation from Poisson process typically less
    than 10 .

33
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

34
Types of coding rate or temporal code?
  • Stimuli that change rapidly tend to generate
    precisely timed spikes

35
Chapter summary
  • Neurons encode information in spike trains
  • Spike rate
  • Time dependent r(t)
  • Spike count r
  • Trial average ltrgt
  • 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

36
Appendix APower spectrum of white noise
  • If Q_ss(t)sigma2 \delta(t) then
    Q_ss(w)sigma2/T
  • Q_ss(w)s(w)2
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