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• 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
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
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
– Arousal, attention
– Randomness in the neuron and
– Other brain processes
Population response
Statistical description
Firing rate
Correlation function
Spike triggered average
Poisson model
Spike trains and firing rates
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
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
• 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
• 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: twospike 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)( 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
• 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  t
Right: rT=10,
The distribution
A Gaussian in n:
Inter-spike interval distribution
• 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:
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  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  t and generate with probability r(t)t, or
– Choose ti+1-tI 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
Experimental validation of Poisson process:
spike counts
Mean spike count and variance of 94 cells (MT macaque) under different
stimulus conditions.
Fit of n2=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:
• 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
– 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
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
• 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