Title: Neurophysics
1Neurophysics
- 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)
2Chapter 1
3Outline
- 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
4Properties of neurons
- Axon, dendrite
- Ion channels
- Membrane rest potential
- Action potential, refractory period
5Synapses, Ca influx, release of neurotransmitter,
opening of post-synaptic channels
6Recording neuronal responses
- Intracellular recording
- Sharp glass electrode or patch electrode
- Typically in vitro
- Extracellular recording
- Typically in vivo
7From 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
8Spike trains and firing rates
9For ? 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
10C Sliding rectangular window D Sliding Gaussian
window
11Causal window
- Temporal averaging with windows is non-causal. A
causal alternative is w(t)a2 t e-a t
E causal window
12Tuning 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
13Motor tuning curves
- Extra cellular recording of monkey primary motor
cortex M1 in arm-reaching task. Average firing
rate is fitted with
14Retinal disparity
- Retinal disparity is location of object on
retina, relative to the fixation point. - Some neurons in V1 are sensitive to disparity.
15Spike-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
16Spike triggered average or reverse correlation
- What is the average stimulus that precedes a
spike?
17Electric fish
- Left electric signal and response of sensory
neuron. - Right C(t)
18Multi-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.
19Spike-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.
20The 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
21Inter-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
22Spike-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
23Autocorrelation for Poisson process
24Inhomogeneous 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
26Poisson 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.
27Model of orientation-selective neuron in V1
- Top orientation of light bar as a function of
time. - Middle Orientation selectivity
- Bottom 5 Poisson spike trials.
28Experimental 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.
29Experimental 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
30Experimental validation of Poisson process
Coefficient of variation
31Shortcomings 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
32Types 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 .
33Types 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
34Types of coding rate or temporal code?
- Stimuli that change rapidly tend to generate
precisely timed spikes
35Chapter 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
36Appendix APower spectrum of white noise
- If Q_ss(t)sigma2 \delta(t) then
Q_ss(w)sigma2/T - Q_ss(w)s(w)2