Unformatted text preview:

Introduction to Neural NetworksU. Minn. Psy 5038Daniel KerstenLecture 2Getting started with MathematicaReview this section in Lecture 1The Neuron - overview of structureFrom Anderson (1995)Basic Structure2 Lect_2_TheNeuron.nbBasic StructureInformation flow: dendrites -> (soma -> axon hillock) -> axon -> terminal buds‡DendritesThe information receiving end of a neuron is a tree-like structure consisting of "dendrites" with special processes or connection sites called synapses. Much computational power is thought to reside in the strength of connections, and in the dendritic tree itself. In this course, we will primarily examine the computational properties of groups of simple neurons, rather than aggregates of dendrites in a single neuron.Dendrites play the role of wires that convey information through changes in voltage. But they behave rather differently than copper wires. These neural processes are tubes of ionized cytoplasm sitting in a bath of ionized fluid whose ionic composition is not that much different from "seawater". The inside of these tubes during resting state sits at about minus 60-70 millivolts relative to the outside of the cell. The tubes are on the order of microns in diameter, (but other processes, such as the axons discussed below, can reach half a millimeter or so). And for further comparison, the membranes making up the tubes are on the order of 50 Angstroms (50 x 10^-10 meters) thick.Signal transmission is limited by high electrical resistance of the axoplasm, and high electrical capacitance of the neural membrane. Information transmission consequences of these properties are: • the voltage potential changes have a short range of influence, with the amplitude decreasing rapidly as one moves away from the synaptic source. • the signals travel relatively slowly.We'll take a quantitative look at these facts shortly.‡Soma (or cell body)• Integrates dendritic signals--spatial integration from sites along a dendrite and between dendrites• The storage of electrical charge across the membrane, and the chemical nature of synaptic transmission leads to temporal integration of signals. This observation together with spatial integration of signals from the dendritic tree arriving at the axon hillock will lead to our basic model of the neuron.Lect_2_TheNeuron.nb 3‡Axon hillock and axonHow can the range and speed be increased? As seen in the above figures, certain neurons are equipped with a specialized process called an axon that serves to "digitize" the data into all-or-none responses (voltage changes) called action potentials or spikes. This digitization occurs at the axon hillock near the cell body. There is passive or electrotonic conduction along the dendrites up to the axon hillock at which point, if there is a sufficient potential change to reach threshold, an active process of depolarization kicks in leading to a spike in membrane voltage. Depolarization means the voltage potential difference across the membrane decreases; hyperpolarization can also occur, where the voltage difference increases.4 Lect_2_TheNeuron.nbThe action potential signals are carried by rapid (1 msec) voltage depolarizations going from -70 to +40 mV via Na+ influx, and K+ outflow through the membrane. From the axon hillock on, a myelin sheath serves to lower the capacitance, increase resistance, and speed up conduction. However it interferes with the regenerative processes that preserve the all-or-none response. At periodic points (Nodes of Ranvier) the myelin sheath is interrupted where high extracellular concen-trations of Na+ ions exist with sodium gates. When a small depolarization arrives, this decreases membrane conductance allowing an increased depolarizing influx of Na+, regenerating the spike.‡Terminal arborization and terminal budsNeurons with axons end in a terminal arborization. The terminal buds make synaptic contacts with the dendrites of subse-quent neurons, and we have the beginnings of a neural network. Synaptic contacts can either be electrical or chemical, but more about these later.Lect_2_TheNeuron.nb 5Basic electrophysiologyPassive propertiesAbove we noted that the potential is maintained by ionic imbalance (excess Na+ outside, and K+ inside). The balance between ionic concentration and electric field forces is determined by the Nernst equation (e.g. see Anderson for a derivation).We noted two problems: passing a signal over a long distance and with sufficient speed--BIG problem for an organism that has to transmit signals fast over a few feet. Let's take a more quantitative look at these problems that arise from the passive electrical properties of neuronal "electronics".Model the passive electrical properties as a function of time :• Across a small portion of the membrane modeled by an "RC circuit" where R stands for resistance, and C for capacitance.We'll see shortly that there is a temporal delay in voltage response characterized by time constant t or rise time.RC-circuits are "low pass temporal filters", i.e. favor signals with low temporal frequencies. and space:• If we imagine cascading a series of RC-circuits, each connected by additional resistances, we have a discrete model of a section of neural membrane. This kind of model is good for computer simulation. But a continuous model can be solved exactly.A continous model over time and short lengths is called the "Cable equation" (Anderson, pages 25-32), made famous by Lord Kelvin (Sir William Thomson, 1824-1907) in the context of submarine telegraph cables (Ireland to Newfoundland in 1858.)We'll see below how the cable equation predicts an exponential drop-off of voltage with distance for constant current. The length constant l ( distance to the 1/e drop-off point or 63% drop) is on the order of millimeters.By solving the cable equation governing the voltage change over distance and time, we can get a quantitative idea of how voltage drops with distance, and how voltage changes with time--the basic message being that change is not instantaneous.6 Lect_2_TheNeuron.nbFrom Segev (1992). A. illustrates an RC-circuit at a single point of passive membrane. B is the temporal response to a step current input. C illustrates additional variable conductance components that model the electrical processes of spike generation (panel D)--the active properties. To model the active properties, one needs a more complicated set of differen-tial equations: the Hodgkin-Huxley equations.


View Full Document
Download The Neuron
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view The Neuron and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view The Neuron 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?