Transcript 48x36 Poster Template - CAAM @ Rice

Perception through interaction of cell assemblies
OPTIONAL
LOGO HERE
OPTIONAL
LOGO HERE
Diane Taylor¹, Steve Cox²
¹Department of Applied Mathematics, Columbia University, New York, NY
²Department of Computational and Applied Mathematics, Rice University, Houston, TX
1. Introduction
Theories of Memory Cont.
Abstract: Cell assemblies are believed by many to be the
fundamental structures in the brain through which we
represent concepts, store and recall information, and form
associations between concepts. Hebb developed a theory
that multiple cell assemblies work together to facilitate each
other’s activity in what he called a phase sequence³. These
phase sequences recall concepts stored in memory due to
previous repeated stimulation, which allows us to perceive
our environment.
•Milner (1957): extended Hebb’s idea of cell assemblies to include
inhibition
•Inhibition: input from one cell to another causes the cell to cease activity;
this is necessary in cell assemblies because otherwise they would continue
to excite themselves forever and cause a seizure6.
•Marr (1971): developed one of the first theories about the storage of
simple memory in the hippocampus through complex interconnected
networks of pyramidal cells5
•His theory of codon formation, first developed for the neocortex, seems to
have a one-to-one correspondence with Hebb’s cell assemblies4
Figure 1
There is no triangle in Figure 1,
but we perceive two because we
have seen so many triangles in
our lives that they have formed a
cell assembly that is recalled by
seeing only the suggestion of
corners. Pattern completion due
to our triangle cell assembly fills
in the rest of the figure for us.
Little is known about how phase sequences work, and Scott
proposed a system of linear homogeneous first order
differential equations (similar to van der Pol) to account for
the inhibition that occur between cell assemblies8. I have
written a Matlab program that simulates the competition
between two cell assemblies.
3. Cell Assemblies
Definition of Cell Assembly:
network of interconnected neurons such that exciting a portion will excite
the whole network, recalling a concept
Hebb’s motivation for cell assemblies:
•Studied patients who had been born blind and then had surgery so that
they ought to be able to see8
•Somehow processing visual perceptions was too difficult for them, so they
preferred to remain blind
•There might be networks that form based on repeated exposure to stimuli
that are essential to perception; blind people would not have been able to
form these networks while they were blind, so they would not be able to
use pattern recognition to process what they saw once the surgery was
performed3
•The networks that process perception must connect several areas of the
brain together
2. A History of Theories of Memory
Cajal (1887): One of the first
scientists to investigate the
microscopic structure of the brain,
made several original and
influential discoveries; notably, the
neuron, able to be observed by
staining the tissue with silver
chromate solution
Figure 2
•Figure 2 shows an actual neuron stained with silver chromate on
the right juxtaposed with one of Cajal’s intricate drawings of a
neuron on the left.
•Lorente de Nó (1938): student of Cajal who noticed that
a cell is fired by simultaneous activity of several afferent cells,
which are arranged in closed circuits
•Afferent cell: takes sensory input and transfers it to the central
nervous system
•Hebb (1949): theory that “cells that fire together wire together”
•Synaptic plasticity: the change in weight of a connection
between neurons due to repeated stimulation; the mechanism by
which cell assemblies form into a cohesive network3
•Was not accepted at the time because he was trying to unite
psychologists and neuroscientists, neither of whom was
interested in the brain on a multi-cellular level8.
Figure 3
Buzsáki’s evidence of cell assemblies:
•Implanted tetrodes in the brains of rats and recorded the firing of neurons
and the position of the rats as they ran around an open environment
collecting food pellets
•His data showed synchronous firing of neurons in the hippocampus
beyond the expected excitation that a neuron would contribute to the
average neuron, if they were not connected by synaptic plasticity in a cell
assembly¹
Palm’s Graph Theoretical Cell Assemblies
•First mathematician to define a cell assemblies as a set of nodes (the
neurons) and edges (the connections between neurons) with specific
properties (exciting a sufficiently large subset would excite the entire
assembly, and it would continually excite itself but none of its neighbors)7
•This concept was helpful for finding cell assemblies within random
networks of cells
•
Figure 4 shows one of Palm’s
•
Figure 4
TEMPLATE DESIGN © 2008
www.PosterPresentations.com
Figure 3 shows Hebb’s drawing of a
simplified cell assembly connecting
Area 17 (the visual cortex) to Area 18.
Cells A and B receive a stimulus, A
excites D and C, D excites E, and C
excites B, and since B has now
received two inputs (the initial stimulus
and the synapse with cell C) it will
propagate the excitation to other parts
of the brain.
1
4. Phase Sequences
Definition of Phase Sequence:
A series of several cell assemblies such that activity of one
facilitates the activity of others
6. MATLAB
MATLAB Program: necker_cube.m
This program simulates perception of a figure that has two
different ways of being perceived.
Figure 7 shows the image
which comes onto the
screen when the program
necker_cube is run. The
Necker cube is a common
optical illusion designed by
Louis Albert Necker in
1832.
Example of Phase Sequence:
Figure 5
Figure 5 shows a triangle broken into three separate parts the way
we perceive it upon receiving the visual stimulus. A possible phase
sequence based on this perception is as follows:
A-B-C-B-A-C-A-T-A-B-T-C-T-B
During the first 7 activations of the subordinate assemblies
(A, B, C), the superordinate structure (T) is being organized.
Activity of T must be transient, and it alternates with
perception of each part of the whole3.
5. Inhibition
The involvement of inhibition within and between cell assemblies
began with Milner’s work because Hebb accepted the belief of his
time that there were only excitatory interactions between cortical
neurons. By Milner’s time, cortical inhibition had been observed,
so he explained this phenomenon with the property that cells
subjected to a constant source of excitation experience a
decrease in frequency of discharge. Each cell will become
fatigued, therefore contributing fewer impulses to the other
neurons in the cell assembly. The inhibitory cells will also fire
less, allowing cells outside the assembly to begin firing, and
eventually the firing of one assembly will stop altogether and
another assembly will take over6.
Scott proposed a system of differential equations to model the
interactions between two cell assemblies similar to a predatorprey model:
Probability of a neuron firing the next time:

 j
I!
F (1  F ) I  j
P( F )   
j   j!( I  j )! 
I
F=fraction of neurons firing within
a cell assembly
I=number of inputs to each neuron
θ=threshold for neuron to give output
Scott’s assumptions: I=2, θ=1
 P( F )  2 F  F 2
F '  ( P( F )  F ) /  where τ is synaptic delay.
This yields the system of ODEs:
dF1
 F1 (1  F1 )  F2
dt
dF2
 F2 (1  F2 )  F1
dt
α is the inhibition constant8.
Figure 6 shows the phase
portrait of the solutions to
this system for the value
α=0.75, which is sufficiently
large
for
individual
assemblies to be ignited.
example networks, which contains 4
cell assemblies: each triangle, the
square, and the entire figure.
Figure 6
Figure 7
The command prompt then asks the user:
“Type the number of the first corner that caught your eye”
The user’s input corresponds to certain initial conditions which
the program uses to solve the differential equations, and it
outputs a plot of what fraction of cells in each assembly are
ignited versus time.
Example: corner 2 is chosen
Figure 8
7. Discussion
Perception is a major aspect of our lives and determines how we react to our
environment in any situation. It can be explained by cell assemblies, which
connect neurons in the visual cortex to other areas of the brain and build
phase sequences to comprehend complex concepts such as using tools and
understanding spoken language. Cell assemblies can be explained to an
extent using graph theory as sets of nodes and edges. This helps to identify
cell assemblies, an important step to further understanding the brain. Cell
assemblies can be more accurately described with mathematics when
inhibition is taken into account, as with Scott’s system of differential equation
for two competing cell assemblies. I have modeled a situation in which cell
assemblies would compete using his equations, but his assumptions for the
threshold and number of inputs are unrealistically low, so I will do further work
on extending his equations to a better model.
8. References
1. Buzsaki, G (2003), Organization of cell assemblies in the hippocampus, Nature, Vol 424, 552 --556.
2. Buzsaki, G (2006), Rhythms of the Brain, Oxford University Press.
3. Hebb, D O (1949) , The Organization of Behavior: a neuropsychological theory, Wileyand Sons
4. Marr, D (1969), A theory of cerebellar cortex, The Journal of Physiology, Vol 202, 437 --470.
5. Marr, D (1971), Simple Memory: A Theory for Archicortex, Philosophical Transactions of the Royal
Society of London, Vol 262, 23 --81.
6. Milner, P M (1957), The Cell Assembly: Mark II, Psychological Review, Vol 64, 242 –252.
7. Palm, G (1981), Towards a Theory of Cell Assemblies, Biological Cybernetics, Vol 39, 181 --194.
8. Scott, A (2002) , Neuroscience: a mathematical primer, Springer-Verlag New York, Inc
Special thanks to Dr. Steve Cox for guidance and mentorship, and to Karina Aliaga, Shaunak Das, and
Tyler Young for consultation.
This work was made possible by Rice University NSF
REU Grant DMS-0755294.