Neural Computation

1
Neural Computation

• Part 1 Neural encoding and decoding (Ch 1-4)
• Part 2 Neurons and Neural circuits (Ch 5-7)
• Part 3 Adaptation and learning (Ch 8-10)

2
Part 1 Neural encoding and decoding

• Stimulus to response (1-2)
• Response to stimulus (3)
• Quantification of information in spikes (4)

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Chapter 1
4
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

5
Properties of neurons

• Axon, dendrite
• Ion channels
• Membrane rest potential
• Action potential, refractory period

6
Synapses, Ca influx, release of neurotransmitter,
opening of post-synaptic channels
7
Recording neuronal responses

• Intracellular recording
• Sharp glass electrode or patch electrode
• Typically in vitro
• Extracellular recording
• Typically in vivo

8
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 D 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)a2 t e-a t

E causal window
13
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

14
Motor tuning curves

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

15
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
  rf(s)x
• Multiplicative noise is proportional to stimulus
  ra s b s x (NB this definition depends on
  stimulus representation).
• 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(t)

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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(t1,t2,,tn)p(t1,t2,
  ,tn)(D 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.

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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
  D t

Right rT10, The distribution Approaches A
Gaussian in n
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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

23
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 ti,ti1 in M segments of
  length D t.
• The probability of no spikes in ti,ti1 is

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• The probability of spikes at times t1,tn is

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Poisson spike generation

• Either
• Choose small bins D t and generate with
  probability r(t)D 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.

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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 sn2A ltngtB yield A,B typically between
  1-1.5, whereas Poisson yields AB1.
• 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
• 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(ti1ti)
• 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 q (7-12 Hz)
  rhythm correlates with location of the animal

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Types of coding 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 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

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