(So you don’t have to watch me draw a lot of bad pictures!) NEURONS! (So you don’t have to watch me draw a lot of bad pictures!) Image from saturn.med.nyu.edu/research/mn/ganlab/

OVERVIEW Neural Modeling Neural network modeling Learning in neural networks Modeling ischemic stroke Self-organizing maps

OVERVIEW CONTINUED Examples A computational tool for the reduction of nonlinear ODE systems possessing multiple scales (Kopell) The dynamic structure underlying subthreshold oscillations and the onset of spikes in medial entorhinal cortex stellate cells (Kopell) Control of repetitive firing in squid axon membrane as a model for a neuron oscillator (Rinzel) Analysis of Competition-Based Spreading Activation in Connectionist Models (Wang) Feature Discovery and Classification of Doppler Umbilical Artery Blood Flow Velocity Waveforms (Baykal) A Connectionist Approach to Vertex Covering Problems (Peng)

OVERVIEW CONTINUED Mathematical Techniques Differential equations Dynamical Systems Computer Science Graph Theory and Networks Analysis

OUR FOCUS Dynamics of the action potential (at a single point) Systems of differential equations: HODGKIN-HUXLEY EQUATIONS Simulation of systems of differential equations (finite difference schemes) Dynamical systems and bifurcations (more generally, some study of excitable systems)

A.L. HODGKIN and A. F. HUXLEY The Nobel Prize in Physiology or Medicine 1963 (with Eccles): "for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the peripheral and central portions of the nerve cell membrane"

More about Hodgkin and Huxley Performed lab experiments on a squid giant axon—very large, up to 1 mm in diameter (normally a micron) Developed voltage clamp experimental technique Developed differential equation model (1 PDE, 3 ODES) for an action potential based on the experiments

5 Publications! Huxley AL and Hodgkin AF. Measurement of Current-Voltage Relations in the Membrane of the Giant Axon of Loligo. Journal of Physiology 1: 424-448, 1952(a). Currents Carried by Sodium and Potassium Ions Through the Membrane of the Giant Axon of Loligo. Journal of Physiology 1:449-472, 1952 (b). The Components of Membrane Conductance in the Giant Axon of Loligo. Journal of Physiology 1: 473-496, 1952 (c). The Dual Effect of Membrane Potential on Sodium Conductance in the Giant Axon of Loligo. Journal of Physiology 1: 497-506,1952 (d). A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. Journal of Physiology 1: 500-544, 1952 (e).

3 weeks on a hand calculator to compute a single action potential!

3 weeks on a hand calculator to compute a single action potential! Your first homework will involve repeating that calculation—just like they did it!

3 weeks on a hand calculator to compute a single action potential! Your first homework will involve repeating that calculation—just like they did it! Just kidding! It is a “repeat” in a sense—but no three week calculations for us!

WHAT NEURONS LOOK LIKE

Many neurons from www.pdn.cam.ac.uk/.../anatomya/opa-neuro.html

Inhibitory Neuron from anatomy.hsc.wvu.edu

Developing cortical neuron from www.anatomy.wisc.edu/faculty_kalil.html

Spiny hippocampal neuron from www.ucihs.uci.edu/anatomy/stewardpix2b.html

Neuron from www.ifisiol.unam.mx/Brain/segunda.htm

Sample nerve cells from www. mind. ilstu. edu/. /neuro/neuron_1 Sample nerve cells from www.mind.ilstu.edu/.../neuro/neuron_1.html (The Mind Project)

BASIC NEURON TYPES 1. 4. 2. 3. 1. Bipolar (interneuron); 2. Unipolar (sensory neuron); 3. Multipolar (motor neuron); 4. Pyramidal Cell (excitatory cortical cell). From www.thethinkingbusiness.co.uk/brainn1.jpg

SOME “PARTS” OF A NEURON

Dendrites (receptors) Take in excitatory & inhibitory stimuli from other neurons or sensory stimuli. Cell Body (processing center) Inside is called the SOMA Image from www.essex.ac.uk/.../sld001.htm

Cell Body Dendrites Axon Hillock Dendrites Axon Image from pespmc1.vub.ac.be/POS/Turchap1.html Axon Synapses

THE AXON Very narrow (microscopic, on the order of a single micron) Very long Human sciatic nerve: nearly 1 meter in length Responsible for action potential propagation!

ACTION POTENTIAL An action potential is a wave of electrical discharge that travels along the membrane of a cell.

Image from wikipedia—under action potential.

Image from wikipedia—under action potential Image from www.learner.org/.../archive/images/1968_d.html

RC Circuit Model Outside of membrane K+ L R-K+ R-L C Resistors: 2 terminal components that resist an electrical current by producing a voltage drop—the axon analogues are the ion channels. The arrow represents a variable resistance (for us, w.r.t. time) Capacitors: Devices that store energy in the electric field created between a pair of conductors on which electrical charges of opposite charge are stored—the axon analogue is created by the membrane. Na+ R-Na+ Inside of Membrane Image from wikipedia under action potential