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 is the probability of any trial firing at time t. B: 100 ms bins

C: Sliding rectangular window D: Sliding Gaussian window

Causal window Temporal averaging with windows is non-causal. A causal alternative is w(t)=[  2 t e -  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)+  –Multiplicative noise is proportional to stimulus r=f(s)  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(  )

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(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.

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:

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

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 [t i,t i+1 ] in M segments of length Δ t. The probability of no spikes in [t i,t i+1 ] is

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

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.

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  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.

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(t i+1 |t i ) 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  (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 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 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