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Modeling Neural Networks

Modeling Neural Networks. Christopher Krycho Advisor: Dr. Eric Abraham May 14, 2009. Computational Neuroscience. Young field (decades old) Modeling the brain with physical principles Oscillatory behaviors Large-scale networks Technique: Create a model of neurons

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Modeling Neural Networks

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  1. Modeling Neural Networks Christopher Krycho Advisor: Dr. Eric Abraham May 14, 2009

  2. Computational Neuroscience • Young field (decades old) • Modeling the brain with physical principles • Oscillatory behaviors • Large-scale networks • Technique: • Create a model of neurons • Model neuron interactions (varying degrees complexity) • Compare results to EEG/MRI/etc. An EEG recording of various locations in a brain

  3. Project Goals • Build a computer program that builds and analyzes small-world networks • Apply that model to neural network models created by the Zochowski group at Michigan • Reproduce neuron and network models • Run a simulation with that model and compare results • Test different models and compare results • Doing new science by examining variations on their model

  4. Outline • What are neurons and neural networks? • Small world networks and their relevance • Modeling a set of neurons in the brain with small world networks • Graph, analyze total synaptic current in system over time.

  5. Neurons • Primary components of nervous system • Transmission and reception of electrical signals • “synaptic” current: current along synapses - connections between neurons Representation of a neuron Source: Wikipedia

  6. The human nervous system Source: http://sciencecity.oupchina.com.hk/biology/ The Nervous System • Composed of neurons throughout • Brain • Nerves in fingers • Spinal cord • 1011 neurons in the brain alone • Each with 105+ connections to other neurons (and itself)

  7. Small World Networks (SWNs) • Small average “path length” • Number of steps from random node to other random node • Large number of nodes • # of nodes in network • Low connectivity • # of connections per node • High clusteredness • Nodes with mutualconnections also connectto each other A network with small world characteristics Source: Exploring Complex Networks, Steven H. Strogatz, Nature410 268-276 (8 March 2001)

  8. Network vs. SWN • “Small world” different from random networks • Despite… • low relative connectivity • Connections primarily short-range • Connections very clustered • … short average path length between any random points in network • SWN architecture observed in brain • On a small scale in local neural networks • Possibly on a larger scale among neural networks

  9. Making SWNs • Initialize an array representing neurons’ connections to each other • Connections initially symmetric within networks • Two or more networks connected internally, but not to each other • Note: actual networks also connected to next-nearest neighbors Two networks of 9 nodes each: A B

  10. From network to SWN, pt. 1 • Randomly rewire the internal connections of each network • Use a random number generator • Connections can be symmetric or directional • Here, symmetric • In model, directional A B The networks after internal rewiring:

  11. From network to SWN, pt. 2 • Randomly add connectivity between networks • Choose some proportion of the neurons in any given network to receive connections • Choose the number in any given network sending connections • Connect - showing directional connections here A The networks after adding connectivity: Key: From A to B From B to A B

  12. Representing Connections • Matrices filled with 0s and 1s • Rows send, columns receive 1 1 1 2 2 1 2 2

  13. A small network rewired • From left to right: • The initial setup of the network - symmetric • The networks, with random rewiring of 0.3 of connections • 25% of neurons on each network receiving input from 4 different neurons on the other network

  14. Representative Connection Graphs • 15x15 grids = 225 neurons/network • 2 networks • 450x450 matrix: • Rows send • Columns receive • Connections to nearest and next-nearest neighbors The networks before any rewiring:

  15. Rep. Conn. Graphs cont’d. The networks after internal rewiring:

  16. Rep. Conn. Graphs cont’d. The networks after adding internetwork connections:

  17. A Neuron Model • Equation representing the state of each neuron at a given time • Numerical value representing charge (V) on the neuron and thus closeness to “spiking” (sending a signal) • Equation involves 4 parameters: • i - Leakage current on ith neuron; constant over time but differs for each neuron • Ji,j(t) - Incoming current to the ith neuron from the jth neurons (connected, in the same network) • Ji,k(t) - Incoming current to the ith neuron from the kth neurons (connected, in another network) • (t) - White noise

  18. Generating Current cont’d. • Equation (identical form for intra/internetwork): • Ji,j(t): current to neuron i from neuron j at time t • Ji,k(t) has the same form, but different lag • s and f ensurecorrect shapeof pulse Current versus time; threshold reached at t=0

  19. Solving the Equation • Integrate over 10 sec with Euler numerical method • Maximum 10-3 sec step size for good resolution (some runs as low as 10-5 sec) • Large computation: • 450 neurons • 5 calculations/neuron/iteration • ~ 225 billion calculations at highest resolution • ~ 45-50 minutes of real time for high resolution • Coded in Fortran • Running on 2007 MacBook Pro

  20. Network Progress

  21. Network Finished

  22. Simulation Progress • Replicating Zochowski model: in progress • Reproducing underlying phenomena • Missing the tell-tale sign of perfectly reproducing their model: “bursting” The Zochowski group’s model graph of network behavior:

  23. Early Simulation Run Network 2 Total synaptic current (arb. Units) Network 1 Time (s)

  24. Present Simulation Run The Zochowski group’s model showing bursting: Our model showing basic behaviors but no bursting

  25. Result Details The Zochowski model individual neurons Our model individual neurons

  26. The Future (of the Project) • Finish replicating Zochowski group’s results • Open doors for the future • Possibility of expanding the number and/or scale of the networks • My design includes • Ability to implement and then test learning mechanisms • Ability to increase complexity of neuron model • Ability to increase complexity of network model

  27. Acknowledgments • The Zochowski group at The University of Michigan, particularly Jane Wang and Sarah Feldt, for their work on which this project is based, and for answering many questions along the way • Dr. Keiran Mullen for teaching me enough Fortran to start the project • Dr. Eric Abraham, my advisor

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