Neurobiology 735 - third module

• Week 1: Coding

• Week 2: Neurons and synapses

• Week 3: Networks

• Week 4: Learning and memory

Textbook: Dayan and Abbott

Coding 1:

Models of neuronal responses to external stimuli

Response of neurons to sensory stimuli

• Average firing rate across trials as a function of time: Post-Stimulus Time Histogram (PSTH)

• Average firing rate in a temporal epoch before the stimulus onset (background rate) or after the

stimulus onset (sensory response)

• Variability across time: Coefficient of Variation (CV) = SD(ISI)/Mean(ISI)

• Variability across trials: Fano Factor (FF) = Var(Spike count)/Mean

CVs of cortical neurons

PFC

Softky and Koch 1993 Compte et al 2003

FFs of cortical neurons

Churchland et al 2010

Modeling spike trains as point processes

I I II I I I

• Renewal processes: each interspike interval is drawn independently from previous

intervals, from a p.d.f. ρ(T ).

Examples of ISI distributions used in neuroscience:

– Exponential (Poisson process)

– Gamma

– Inverse Gaussian

Example 1: Poisson

• pdf of ISIs

P (T ) = ν exp(−νT )

• mean

1/〈T 〉 = ν

• CVSD(T )

〈T 〉= 1

• Spike counts in disjoint intervals are in-

dependent.

• Spike count in an interval of duration T

is given by a Poisson distribution,

P (k) = (νT )k exp(−νT )/k!

0 1 2 3 4 5Interspike Interval (ISI)/Mean

0

0.2

0.4

0.6

0.8

1

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• Mean spike count in interval of duration

T : νT

• Variance of spike counts is equal to the

mean

• Fano Factor FF=1

Example 2: Gamma

• pdf of ISIs

P (T ) =(kν)k

Γ(k)T k−1 exp(−kνT )

• mean

1/〈T 〉 = ν

• CVSD(T )

〈T 〉=

1√k

0 1 2 3 4 5Interspike Interval (ISI)/Mean

0

0.2

0.4

0.6

0.8

1

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k=0.5

k=1

k=2

k=5

Example 3: Inverse Gaussian

• pdf of ISIs

P (T ) =

√k

2πT 3νexp

(−k(νT − 1)2

2Tν

)• mean

1/〈T 〉 = ν

• CVSD(T )

〈T 〉=

1√k

0 1 2 3 4 5Interspike Interval (ISI)/Mean

0

0.5

1

1.5

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sity

k=0.5

k=1

k=2

k=5

How do spike trains depend on external stimuli?

1. Mean rates

Continuous stimuli

Tuning curves quantify how the firing rate of a neuron depends on a continuous parameter

characterizing the stimulus.

• Bell-shaped tuning curves

– Orientation selectivity in V1;

– Direction selectivity in MT;

– Spatial location of the animal in HPC of the rat;

– Spatial location of stimulus in PPC, PFC;

– Location of a saccade in FEF;

– Direction of the arm in M1;

– Head direction in DTN, thalamus, subiculum;

• Monotonic tuning curves

– Eye position in oculomotor nuclei

– Angular velocity of the head in vestibular nuclei

– Frequency of vibration in S1, S2, PFC

– Retinal disparity in V1

V1: orientation

r = r0 + (rmax − r0) exp

(−1

2

(s− smax

σ

)2)

Hubel and Wiesel 1968

M1: arm direction

r = r0 + (rmax − r0) cos(s− smax)

Georgopoulos et al 1982

Auditory system: frequency

Bartlett et al 2011

Prefrontal cortex

r = r0 + (rmax − r0) exp

(−1

2

(s− smax

σ

)2)

Funahashi et al 1989

Head-direction cells

Taube 1995

Oculomotor nuclei: eye position

Aksay et al 2000

Discrete stimuli

• Olfactory system: odors

• Temporal lobe: objects

Typically, neurons respond only to a small fraction of all possible stimuli (sparse coding)

Rat olfactory cortex - odors

Poo and Isaacson 2009

Primate IT cortex - objects

Woloszyn and Sheinberg 2012

Human hippocampus - people

Quiroga et al 2005

How are dynamic stimuli encoded by single neurons?

Characterizing input/output transformation: Volterra/Wiener series

• How to characterize the transformation from a time-dependent input s(t) into a

time-dependent neuronal output r(t)?

• Volterra/Wiener series:

r(t) = F [s(t)] = g0 +

∫dt1g1(t1)s(t− t1)

+

∫dt1dt2g2(t1, t2)s(t− t1)s(t− t2)

+ . . .

• gn = Volterra/Wiener kernels

Wiener series

• Wiener approach:

– Use a stochastic signal s(t) (simplest case: white noise)

– In the Wiener expansion, individual terms are statistically independent

Approximating white noise

• White noise = mathematical idealization

• No physical system can generate truly white

noise

• Approximation: generate noise at discrete time

steps, t = m∆t where m is an integer, ∆t

should be much smaller than the characteristic

time scales of the system;

• At each time step, generate s from a Gaussian

pdf with mean zero and variance S/∆t, inde-

pendently from previous steps

Computing Wiener kernels

• Use white noise stimulus s(t)

• Record spike train y(t) in response to the stimulus:

I I II I I I

• Correlate output with products of white noise input at different times::

g0 = 〈y(t)〉g1(τ) = 〈y(t)s(t− τ)〉

g2(τ1, τ2) = 〈y(t)s(t− τ1)s(t− τ2)〉. . .

• g0 = mean firing rate

• g1(τ) = Spike-Triggered Average of the stimulus

• g2(τ1, τ2) = Spike-Triggered Covariance of the stimulus

• . . .

The spike-triggered average (STA)

• Stimulus s(t)

• Measure spike train

y(t) =

n∑i=1

δ(t− ti)

• Spike triggered average (STA)

g1(τ) =1

n

n∑i=1

s(ti − τ)

=

∫s(t− τ)y(t)dt∫

y(t)dt

Visual system: Spatio-temporal receptive fields

• Spatio-temporal white-noise stimulus

s(x, y, t)

• Compute first-order Wiener kernel

g(x, y, τ) using STA

• Receptive field = {x, y} for which

g(x, y, τ) 6= 0

• Separable receptive field:

g(x, y, τ) = gs(x, y)gt(τ)

• Non-separable receptive field:

g(x, y, τ) 6= gs(x, y)gt(τ)

RFs in thalamus and retina

• Circular central ON (OFF) surrounded by annu-

lar OFF (ON)

• Fitted by a difference of Gaussians:

gs(x, y) = ±(

1

2πσ2c

exp

(−x

2 + y2

2σ2c

)− B

2πσ2s

exp

(−x

2 + y2

2σ2s

))– B = balance between center and surround

– σc = width of center

– σs = width of surround

Spatial RF of a simple cell in V1

Fitted by a Gabor function:

gs(x, y) = exp

(−x′2 + γ2y′2

2σ2

)cos

(2πx′

λ+ ψ

)x′ = x cos(θ) + y sin(θ)

y′ = −x sin(θ) + y cos(θ)

• λ = wavelength

• θ = orientation

• ψ = phase offset

• σ = width of Gaussian envelope

• γ = spatial aspect ratio

Temporal evolution of a spatial receptive field

Temporal structure of a RF

LNP model

• Can we account for non-linearities in a simple way?

• Linear-Nonlinear-Poisson (LNP) neuron model:

– Instantaneous firing rate given by

r(t) = Φ

(∫ t

−∞g1(t− t′)s(t′)dt′

)characterized by temporal filter (kernel) g1, and static non-linearity Φ

– Spike train = Poisson process with instantaneous firing rate r(t)

Fitting a LNP model to data

• Instantaneous firing rate given by

r(t) = Φ

(∫ t

−∞g1(t− t′)s(t′)dt′

)• Compute temporal filter g1 from the data

using STA;

• Then plot r(t) as a function of

L =

∫ t

−∞g1(t− t′)s(t′)dt′

• Compute Φ as the average of the data

points, and (optional) fit it with a particular

function.

Example: fitting retinal ganglion cell data (Pillow et al 2005)

Model vs data

Bibliography

• Dayan and Abbott, “Theoretical Neuroscience: Computational and Mathematical

Modeling of Neural Systems” (MIT Press, 2001), chapters 1& 2