Brief Notes on Theoretical Neuroscience
April 14, 2010
Gabriel Kreiman
http://klab.tch.harvard.edu
Outline
1.
2.
3.
4.
5.
Why theoretical neuroscience?
Single neuron models
Network models
Algorithms and methods for data analysis
A sample of a few computational studies
(visual recognition)
Further reading
This will NOT be an exhaustive presentation of work in Theoretical
Neuroscience…
•Abbott and Dayan. Theoretical Neuroscience - Computational and
Mathematical Modeling of Neural Systems [2001] (ISBN 0-26204199-5). MIT Press.
•Koch. Biophysics of computation [1999] (ISBN 0-19-510491-9).
Oxford University Press.
•Hertz, Krogh, and Palmer, Introduction to the theory of neural
computation. [1991] (ISBN 0-20151560-1). Santa Fe Institute Studies
in the Sciences of Complexity.
Why bother with models?
-Quantitative models force us to think about and formalize hypotheses and
assumptions
-Models can integrate and summarize observations across experiments, resolutions
and laboratories
-A good model can lead to (non-intuitive) experimental predictions
-A quantitative model, implemented through simulations, can be useful from an
engineering viewpoint (e.g. face recognition)
-A model can point to important missing data, critical information and decisive
experiments
Do I have to be a “professional theoretician” to build
a model?
-No!
-There are plenty
experimentalists…
of
excellent
computational
papers
written
by
A very brief sample:
-Laurent G (2002) Olfactory Network Dynamics and the coding of multidimensional signals.
Nature
Reviews
Neuroscience
3:884-895.
- Carandini M, Heeger DJ, Movshon JA (1997) Linearity and normalization in simple cells of
the
macaque
primary
visual
cortex.
J
Neurosci
17:8621-8644.
- Brincat S, Connor C (2006) Dynamic Shape Synthesis in Posterior Inferior Temporal
Cortex.
Neuron
49:17-24
- Blake R (1989) A neural theory of binocular rivalry. Psychological Review 96:145-167.
- Hubel, D.H. and T.N. Wiesel, Receptive fields, binocular interaction and functional
architecture in the cat's visual cortex. J Physiol, 1962. 160:106-54.
-Scientists always build models (even if the models are not quantitative and
are not implemented through computational simulations). There is no such
thing as a “model-free” experiment…
A model for orientation tuning in simple cells
A feed-forward model for orientation
selectivity in V1
(by no means the only model)
Hubel and Wiesel. J. Physiology (1962)
More anatomical complexity – Similar math
Douglas and Martin 2004
Felleman and Van Essen 1991
Outline
1.
2.
3.
4.
5.
Why theoretical neuroscience?
Single neuron models
Network models
Algorithms and methods
A sample of a few computational studies
A nested family of single neuron models
Filter
operations
Integrateand-fire
circuit
HodgkinHuxley units
Multicompartmental
models
Biological
Spines,
accuracy
channels
Lack of analytical
solutions
Computational
complexity
Geometrically accurate models vs. spherical cows
with point masses
A central question in Theoretical Neuroscience:
What is the “right” level of abstraction?
The leaky integrate-and-fire model
•
•
•
•
•
•
Lapicque 1907
Below threshold, the voltage is
governed by:
dV(t)
V(t)
C

 I(t)
dt
R
A spike is fired when V(t)>Vthr (and
V(t) is reset)
A refractory period tref is imposed
after a spike.
Simple and fast.
Does not consider spike-rate
adaptation, multiple
compartments, sub-ms biophysics,
neuronal geometry
Vrest=-65 mV
Vth =-50 mV
Τm = 10 ms
Rm = 10 MΩ
Line = I&F model
Circles = cortex
first 2
spikes
adapted
The Hodgkin-Huxley Model
dV
I (t )  C
 g L (V  EL )  g K n 4 (V  EK )  g Na m3h(V  E Na )
dt
where:
im = membrane current
V = voltage
L = leak channel
K = potassium channel
Na = sodium channel
g = conductances (e.g. gNa=120 mS/cm2; gK=36 mS/cm2; gL=0.3 mS/cm2)
E = reversal potentials (e.g. ENa=115mV, EK=-12 mV, EL = 10.6 mV)
n, m, h = “gating variables”, n=n(t), m=m(t), h=h(t)
Hodgkin, A. L., and Huxley, A. F. (1952).
A quantitative description of membrane current and its application to conduction and excitation in nerve.
Journal of Physiology 117, 500-544.
Outline
1.
2.
3.
4.
5.
Why theoretical neuroscience?
Single neuron models
Network models
Algorithms and methods
A sample of a few computational studies
From neurons to circuits
•Single neurons can perform many interesting and important
computations (e.g. Gabbiani et al (2002). Multiplicative computation in a visual neuron
sensitive to looming. Nature 420, 320-324)
•Neurons are not isolated. They are part of circuits. A typical
cortical neuron receives input from ~104 other neurons.
•It is not always trivial to predict circuit-level properties from single
neuron properties. There could be interesting properties emerging
at the network level.
Circuits – some basic definitions
Notes:
1.
Connectivity does not need to be all-to-all
2.
There are excitatory neurons and inhibitory neurons (and many types of inhibitory
neurons)
3.
Most models assume balance between excitation and inhibition
4.
Most models do not include layers and the anatomical separation of forward and back
pathways
5.
There are many more recurrent+feedback connections than feed-forward connections
(the opposite is true about models…)
Firing rate network models – A simple feedforward circuit
• Time scales > ~ 1 ms
• Analytic calculations in
some cases
• Fewer free parameters
than spiking models
N
t
I

w
• Easier/faster to
 b  dK s (t   )ub ( )
s
b1
simulate

N
dIs
s
 Is   w b ub
dt
b1
v  F (I s )

if

Is = total synaptic current
N = total number of inputs
wb = synaptic weights
Ks(t) = synaptic kernel
ub = input firing rates
K s (t)  (1/  s )exp( t /  s )
F can be a sigmoid function
Or a threshold linear function:
F ( I s )  [ I s   ]
Learning from examples – The perceptron
Imagine that we want to
classify the inputs u into two
groups “+1” and “-1”
Training examples: {um,vm}
w w


Ú
vm  v(um ) um
2
Perceptron learning rule
Linear separability: can attain zero error
Cross-validation: use separate training and test data
There are several more sophisticated learning algorithms
Learning from examples – Gradient descent
Now imagine that v is a real
value (as opposed to binary)
u  f (s)
v(s)  w.u
We want to choose the weights
so that the output approximates
some function h(s)
Move along the gradient of the error between the desired output and
the current output
Outline
1.
2.
3.
4.
5.
Why theoretical neuroscience?
Single neuron models
Network models
Algorithms and methods
A sample of a few computational studies
Some examples of computational algorithms and methods
• Different techniques for time-frequency analysis of neural signals
(e.g. Pesaran et al 2002, Fries et al 2001)
• Spike sorting (e.g. Lewicki 1998, Quian Quiroga et al 2005)
• Machine learning approaches to decoding neuronal responses
(e.g. Hung et al 2005, Wilson et al 1993, Musallam et al 2004)
• Information theory (e.g. Abbott et al 1996, Bialek et al 1991)
• Neural coding (e.g. Gabbiani et al 1998, Bialek et al 1991)
• Definition of spatio-temporal receptive fields, phenomenological
models, measures of neuronal synchrony, spike train statistics
Outline
1.
2.
3.
4.
5.
Why theoretical neuroscience?
Single neuron models
Network models
Algorithms and methods
A sample of a few computational studies
Predicting spikes in the retina
g(t)  s(t)* F(t)

s(t) = visual stimulus
F(t) = linear filter
“*” = convolution
g(t)= “generator” potential
= threshold
r(t) = firing event (1 when g(t)>)
Keat J, Reinagel P, Reid RC, Meister M (2001) Predicting every spike: a model for the responses
of visual neurons. Neuron 30:803-817.
Predicting spikes in the retina
h(t)=g(t)+P(t)
P(t)=Bexp(-t/)
Keat J, Reinagel P, Reid RC, Meister M (2001) Predicting every spike: a model for the responses
of visual neurons. Neuron 30:803-817.
Predicting spikes in the retina
Two “noise” sources are added
to account for trial-to-trial
variability:
a(t) = gaussian noise at the
generator potential
b(t) = random fluctuations in
the feedback potential P(t)
Keat J, Reinagel P, Reid RC, Meister M (2001) Predicting every spike: a model for the responses
of visual neurons. Neuron 30:803-817.
The “blue brain” modeling project
-http://bluebrain.epfl.ch
- IBM’s Blue gene supercomputer
- “Reverse engineer” the brain in a “biologically accurate” way
- November 2007 milestone: 30 million synapses in “precise” locations to model
a neocortical column
- Compartmental simulations for neurons
- Needs another supercomputer for visualization (10,000 neurons, high quality
mesh, 1 billion triangles, 100 Gb)
QUESTION: What is the “right” level of abstraction needed to understand
the function of cortical circuitry?
An object can cast an infinite number of projections on the retina
A brute force approach to object recognition
Task: Recognize the
handwritten “A”
A “brute force” solution:
- Use templates for each letter
- Use multiple scales for each
template
- Use multiple positions for
each template
- Use multiple rotations for
each template
- Etc.
Problems with this approach:
- Large amount of storage for
each object
- No extrapolation, no
intelligent learning
- Need to learn about each
object under each
condition
A non-exhaustive list of biological object recognition models
Some common themes across multiple models:
•Hierarchical structure
•Increased “receptive field” size
•Increased complexity in shape preferences
•Increased invariance
K. Fukushima, Neocognitron: a self organizing neural network model for a mechanism of pattern recognition
unaffected by shift in position. Biological Cybernetics, 1980. 36: 193-202.
Y. LeCun, L. Bottou, Y. Bengioand P. Haffner, Gradient-based learning applied to document recognition.
Proc of the IEEE, 1998. 86: 2278-2324.
G. Wallis and E.T. Rolls, Invariant face and object recognition in the visual system. Progress in
Neurobiology, 1997. 51: 167-94.
B. Mel, SEEMORE: Combining color, shape and texture histogramming in a neurally inspired approach to
visual object recognition. Neural Computation, 1997. 9: 777.
B.A. Olshausen, C.H. Anderson and D.C. Van Essen, A neurobiological model of visual attention and
invariant pattern recognition based on dynamic routing of information. J Neurosci, 1993. 13: 4700-19.
M. Riesenhuberand T. Poggio, Hierarchical models of object recognition in cortex. Nature Neuroscience,
1999. 2: 1019-1025.
G. Deco and E.T. Rolls, A neurodynamical cortical model of visual attention and invariant object recognition.
Vision Res, 2004. 44: 621-42.
P. Foldiak, Learning Invariance from Transformation Sequences. Neural Computation, 1991. 3: 194-200.
Neocognitron
Retinotopically arranged connections between layers
Feature extracting “S” cells
C-cells performing a local “OR” operation
Increasing buildup of position tolerance
Unsupervised learning in S layers
Fukushima K. (1980) Neocognitron: a self organizing neural network model for a mechanism
fo pattern recognition unaffected by shift in position. Biological Cybernetics 36, 193-202
Recognizing objects by part decomposition
Biederman (1987) Psychological Review
Object recognition by alignment to prototypes
Prototype
Alignment of 3 points to the prototype (black arrows)
Note: some points may not align (red ellipses)
Ullman (1996) High-level vision
Invariance in visual object recognition
Several computational models for rotation invariance rely on building 3D object models
Here, recognition is based on learning from few perspective views
Generalized radial basis functions
K
f (x)   ciG(|| x  t i ||)
i 1
ti = centers
ci = coefficients
G = basis function (e.g. gaussian)
Poggio T, Edelman S (1990) A network that learns to recognize 3D objects. Nature 343:263-266.
Invariance in visual object recognition
Poggio T, Edelman S (1990) A network that learns to recognize 3D objects. Nature 343:263-266.
Computer vision
Wang et al CVPR 2006
- Goal: “generic” object
recognition
- Challenge: object
transformations, intraclass variability for
categorization
- e.g. Caltech 101
dataset, 30-800
exemplars/category
•Learning generative visual models.L. Fei-Fei, R. Fergus, and P. Perona. CVPR 2004
Shape Matching and Object Recognition using Low Distortion Correspondence.
Alexander C. Berg, Tamara L. Berg, JitendraMalik. CVPR 2005
•The Pyramid Match Kernel:DiscriminativeClassification with Sets of Image Features.
K. Graumanand T. Darrell. ICCV) 2005.
•Combining Generative Models and Fisher Kernels Holub, AD. Welling, M. Perona, P.
ICCV 2005
•Exploiting Unlabelled Data for Hybrid Object Classification.Holub, AD. Welling, M.
Perona, P. NIPS 2005 Workshop in Inter-Class Transfer.
•Object Recognition with Features Inspired by Visual Cortex. T. Serre, L. Wolf and T.
Poggio. CVPR 2005.