1. TEMPLATE DESIGN © 2008 www.PosterPresentations.com Perception through interaction of cell assemblies Diane Taylor ¹, Steve Cox ² ¹ Department of Applied Mathematics, Columbia University, New York, NY ² Department of Computational and Applied Mathematics, Rice University, Houston, TX 1. Introduction 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 sequence ³. These phase sequences recall concepts stored in memory due to previous repeated stimulation, which allows us to perceive our environment. 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 assemblies 8. I have written a Matlab program that simulates the competition between two cell assemblies. 2. A History of Theories of Memory 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 see 8 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 performed 3 The networks that process perception must connect several areas of the brain together 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 assembly ¹ 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 1 Theories of Memory Cont.4. Phase Sequences 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 over 6. Scott proposed a system of differential equations to model the interactions between two cell assemblies similar to a predator- prey model: 6. MATLAB 7. Discussion 8. References MATLAB Program: necker_cube.m This program simulates perception of a figure that has two different ways of being perceived. 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. 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. 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 network 3 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 level 8. OPTIONAL LOGO HERE Figure 1 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 Definition of Phase Sequence: A series of several cell assemblies such that activity of one facilitates the activity of others Example of Phase Sequence: 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 whole 3. 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 seizure 6. Marr (1971): developed one of the first theories about the storage of simple memory in the hippocampus through complex interconnected networks of pyramidal cells 5 His theory of codon formation, first developed for the neocortex, seems to have a one-to-one correspondence with Hebb’s cell assemblies 4 Figure 2 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. Figure 5 Figure 3 α is the inhibition constant 8. 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. Figure 6 Figure 7 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. 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 Probability of a neuron firing the next time: F=fraction of neurons firing within a cell assembly Scott’s assumptions: I =2, θ =1 I =number of inputs to each neuron θ =threshold for neuron to give output where τ is synaptic delay. This yields the system of ODEs: Figure 8 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. Figure 4 Figure 4 shows one of Palm’s example networks, which contains 4 cell assemblies: each triangle, the square, and the entire figure.