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)

Chapter 1

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

Properties of neurons

Axon, dendrite

Ion channels

Membrane rest potential

Action potential, refractory period

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

Recording neuronal responses

• Intracellular recording– Sharp glass electrode or

patch electrode

– Typically in vitro

• Extracellular recording– Typically in vivo

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

Spike trains and firing rates

For Δ t ! 0, each interval contains 0,1 spike. Then, r(t) averaged over trials isthe probability of any trial firing at time t.

B: 100 ms bins

C: Sliding rectangular windowD: Sliding Gaussian window

Causal window

• Temporal averaging with windows is non-causal. A causal alternative is w(t)=[α2 t e-α t]+

E: causal window

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

Motor tuning curves

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

Retinal disparity

• Retinal disparity is location of object on retina, relative to the fixation point.

• Some neurons in V1 are sensitive to disparity.

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)+ξ– Multiplicative noise is proportional to stimulus r=f(s) ξ

• statistical description– Spike triggered average– Correlations

Spike triggered average or reverse correlation

• What is the average stimulus that precedes a spike?

Electric fish

• Left: electric signal and response of sensory neuron.• Right: C(τ)

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.

Spike-train statistics

• If spikes are described as stochastic events, we call this a point process: P(t1,t2,…,tn)=p(t1,t2,…,tn)(Δ t)n

• The probability of a spike can in principle depend on the whole history: P(tn|t1,…,tn-1)

• If the probability of a spike only depends on the time of the last spike, P(tn|t1,…,tn-1)=P(tn|tn-1) it is called a renewal process.

• If the probability of a spike is independent of the history, P(tn|t1,…,tn-1)=P(tn), it is called a Poisson process.

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 distributionApproaches A Gaussian in n:

• Suppose a spike occurs at tI, what is the probability that the next spike occurs at tI+1?

• Mean inter-spike interval:

• Variance:

• Coefficient of variation:

Inter-spike interval distribution

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

Autocorrelation for Poisson process

Inhomogeneous Poisson Process

• Divide the interval [ti,ti+1] in M segments of length Δ t.

• The probability of no spikes in [ti,ti+1] is

• The probability of spikes at times t1,…tn is:

Poisson spike generation

• Either– Choose small bins Δ t and generate with probability r(t)Δ t, or

– Choose ti+1-tI from p(τ)=r exp(-r τ)

• Second method is much faster, but works for homogeneous Poisson processes only

• It is further discussed in an exercise.

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.

Experimental validation of Poisson process: spike counts

• Mean spike count and variance of 94 cells (MT macaque) under different stimulus conditions.

• Fit of σn2=A <n>B yield A,B typically between 1-1.5, whereas Poisson

yields A=B=1.• variance higher than normal due to anesthesia.

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

Experimental validation of Poisson process: Coefficient of variation

• MT and V1 macaque.

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

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(ti+1|ti)

• Deviation from Poisson process typically less than 10 %.

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 θ (7-12 Hz) rhythm correlates with location of the animal

Types of coding: rate or temporal code?

• Stimuli that change rapidly tend to generate precisely timed spikes

Chapter summary

• Neurons encode information in spike trains• Spike rate

– Time dependent r(t)

– Spike count r

– Trial average <r>

• 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

Appendix APower 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