Department of Mathematical Sciences B12412: Computational Neuroscience and Neuroinformatics Why use models? Successes in Mathematical Neuroscience. 1950's The 1952 work of Alan Hodgkin & Andrew Huxley model led to the award of the 1963 Nobel Prize in Physiology or Medicine (together with Sir John Eccles) for their work on understanding nerve action potentials in the squid giant axon. The experimental measurements on which the pair based their action potential theory represent one of the earliest applications of a technique of voltage clamp. The second critical element of their research was the so-called giant axon of the Atlantic squid (Loligo pealei). The large diameter of the axon (up to 1 mm in diameter; typically around 0:5 mm) provided a great experimental advantage for Hodgkin and Huxley as it allowed them to insert voltage clamp electrodes inside the lumen of the axon. The pair published their theory in 1952. In the paper, they describe one of the earliest computational models in biochemistry, that is the basis of most of the models used in Neurobiology during the following four decades. In the course of their research, Huxley and Hodgkin were surprised The giant axon of the squid, not the axon of the giant squid! to learn that, contrary to earlier hypotheses, the outer layer of a nerve fiber is not equally permeable to all ions (charged particles). While a resting cell has low sodium- and high potassium- permeability, Huxley and Hodgkin found that, during excitation, sodium ions flood into the axon, which instantaneously changes from a negative to a positive charge. It is this sudden change that constitutes a nerve impulse. The sodium ions then continue to flow through the membrane until the axon is so highly charged that the sodium becomes electrically repelled. The stream of sodium then stops, which causes the membrane to become permeable once again to potassium ions. Action potentials 80 0 m/s The Hodgkin-Huxley model of an axon is now widely recognised as a model of excitable media - 1 Basic Components of Hodgkin-Huxley type Models. Hodgkin-Huxley type models represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (Cm). Voltage-gated and leak ion channels are represented by nonlinear (gn) and linear (gL) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (Ip). at every point along an axon a patch of membrane is described by a set of four nonlinear ordinary differential equations. Activity spreads in a regenerative way along the axon via diffusive coupling between neighbouring patches of excitable tissue. The form of the action potential is predicted extremely well by the model. For this, integration of the differential equations was required. Huxley devised an algorithm for use on a mechanical hand calculator. Each run of the algorithm producing a 5 millisecond theoretical voltage trace took about 8 hours of effort { Quote from Hodgkin: The propagated action potential took about three weeks to complete and must have been an enormous labour for Andrew [Huxley]. A "Millionaire" mechanical calculator. For a wonderful review of the Hodgkin-Huxley framework see the paper by John Rinzel1 For the past forty years our understanding and our methods of studying the biophysics of excitable membranes have been significantly influenced by the landmark work of Hodgkin and Huxley. Their 11990, Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and an update, Bulletin of Mathematical Biology, Volume 52, 3-23. 2 work has had far reaching impact on many different life science subdisciplines where concepts of cell biology have come to be important. These include not only neurophysiology but also endocrinology, muscle and cardiac physiology, and developmental biology. The ionic currents and electrical signals generated by neuronal membranes are of obvious importance in the nervous system. But ionic fluxes also play important roles in affecting cellular functions such as secretion, contraction, migration, etc. Voltage-dependent conductances like those which Hodgkin and Huxley discovered, and the ionic channels which they imagined (and which are now known to exist), are ubiquitous in the cells of animals and plants. A number of java applets showing the dynamics of excitable media (in both neural and cardiac) contexts can be found online. See for example the model of an action potential at http://lcn.epfl.ch/tutorial/english/hodhux/html/axon.html Numerous computations and analytic studies have now been performed with Hodgkin-Huxley like equations to account for the rich repertoire of nonlinear phenomena in excitable membranes, as well as to uncover new behaviors in the models, and to suggest further experiments. In example, the figure below shows the result of a mathematical analysis of travelling periodic wave speed (multiple spikes) in the Hodgkin-Huxley model. 1960's In the 1960s Rall developed the cable model of the dendritic tree. Dendrites are strikingly exquisite and unique structures. They are the largest component in both surface area and volume of the brain and their specific morphology is used to classify neurons into classes: pyramidal, Purkinje, amacrine, stellate, etc. Cable theory describes how membrane potential spreads along the dendritic branches in response to a local conductance change (synaptic input) In the last thirty years, cable theory for dendrites, complemented by the compartmental modeling approach has played an essential role in the estimation of dendritic parameters, in designing and interpreting experiments and in providing insights into the computational function of dendrites. Cable theory Dendrites are thin tubes wrapped with a membrane that is a relatively good electrical insulator compared to the resistance provided by the intracellular core or the extracellular fluid. Because 3 Left: Computed dispersion relations, conduction speed vs. firing frequency, for periodic traveling wave solutions to the Hodgkin-Huxley equations for three temperatures. Speeds of the fast and slow solitary pulses (shown dashed) represent the limiting speeds of the periodic waves as the frequency approaches zero. Right: The relation between conduction speed and wavelength for traveling wave solutions to the Hodgkin-Huxley equations. From R N Miller and J Rinzel Biophys J. 1981 May; 34(2): 227259. The dependence of impulse propagation speed on firing frequency, dispersion, for the Hodgkin-Huxley model. of this difference in membrane vs. axial resistivity, for a short length of dendrite, the electrical current inside the core conductor tends to flow parallel to the cylinder axis (along the x-axis). Cable theory considers dendritic segments to have just one spatial dimension (x) The other fundamental assumptions in cable theory are: 1. The membrane is passive (voltage-independent) and uniform. 2. The core conductor has constant cross section and the intracellular fluid can be represented as an ohmic resistance. 3. The extracellular resistivity is negligible (implying extracellular isopotentiality). Importantly the cable equation can be solved analytically for arbitrarily complicated passive den- dritic trees. The cable equation is a second order partial differentiation equation @V V 1 @2V C = - + ; 2 @t R ra @x where R is the membrane resistivity, ra is the cytoplasm resistivity and C the membrane capaci- tance for a unit length of cable with fixed diameter. An equivalent cylinder Rall argued that when a cylinder with diameter dp bifurcates into two daughter branches with diameters d1 and d2 , the branch point behaves as a continuous cable for current that flows from the parent to daughters, if 3=2 3=2 3=2 dp = d1 + d2 A branch point obeying Rall's law is electrically equivalent to a uniform cylinder (looking from the parent into the daughters). Rall extended this concept (impdeance matching) to trees and showed that (from the soma viewpoint out to the dendrites) there is a subclass of trees that are electrically equivalent to a single cylinder whose diameter is that of the stem (near the soma) dendrite. Dendrites of many neuron types (e.g., some α-motoneuron) obey, to a first approximation, the 3=2 rule. However, the dendrites of several major types of neurons (e.g., 4 A Purkinje cell. cortical and hippocampal pyramidal cells) do not obey this rule. Still, the equivalent cylinder model for dendritic trees allows for a simple analytical solution and, indeed, it has provided the main insights regarding the spread of electrical signals in passive dendritic trees. Compartmental modelling The compartmental modeling approach complements cable theory by overcoming the assumption that the membrane is passive. Mathematically, the compartmental approach is a finite-difference (discrete) approximation to the (nonlinear) cable equation. It replaces the continuous cable equation by a set, or a matrix, of ordinary differential equations and, typically, numerical methods are employed to solve this system (which can include thousands of compartments and thus thousands of equations) for each time step. Conceptually, in the compartmental model dendritic segments that are electrically short are assumed to be isopotential and are lumped into a single RC (either passive or active) membrane compartment. Compartments are connected to each other via a longitudinal resistivity according to the topology of the tree. Hence, differences in physical properties (e.g., diameter, membrane properties, etc.) and differences in potential occur between compartments rather than within them. It can be shown that when the dendritic tree is divided into sufficiently small segments (compartments) the solution of the compartmental model converges to that of the continuous cable model. A compartment can represent a patch of membrane with a variety of voltage-gated (excitable) and synaptic (time-varying) channels. 1970's The 1970s saw the development of neural field equations2. These are tissue level models that describe the spatio-temporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons.