Neurophysics

Figure 17: Different levels of discretizing and simplifying neuronal structure as a first step in solving numerically the membrane potential dynamics. The size of the compartments is chosen such that the membrane potential is approximately the same over the size of the compartment. Compartments are connected to each other via resistive couplings.

In a multi-compartment model, each compartment has its own membrane potential Vμ ( μ labelling the compartment) and its own membrane current imμ depending on the composition of ion-channels

within the compartment. In the case of a single independent compartment, the membrane potential is described by:

with cm being the specific membrane capacitance per unit area (1 μF/cm2), A the compartment surface area, Ie an externally injected current via e.g. a patch-clamp electrode or a synapse and im describing the sum of membrane currents flowing through ion-channels.

When several compartments are coupled in a non-branching manner, the above equation is modified to include the currents exchanged by the compartments:

with Ieμ being the total electrode current flowing into compartment μ and Aμ being the surface area

of the compartment. In the case when the compartment is at the end of a cable, having only one neighbour, a single term will replace the last two terms in the above equation. For a compartment where a cable branches in two, there will be three such terms, corresponding to coupling of the branching node to the first compartment in each of the daughter branches. The compartments are coupled through terms containing constants (conductances) of the type gμ,μ' that determine the

strength of resistive Ohmic coupling between compartments μ and μ' . If the compartments μ and μ' have the same length L and radius a then:

The above equation is obtained by using the approximation that for small enough discretization, the capacitance and ion-channels within a compartment are located at its center and that the centers of

two compartments are a distance L apart. Like this, the resistance between the centers of the two compartments is:

with rL being the intracellular resistivity and a the radius of the compartments. Since the currents on the right hand side of eq. (36) are defined per compartment area, eq. (38) has to be divided by 2πaL and finally inverted to yield eq. (37). In the more general case when the lengths of the compartments and the radii may be different, the coupling constants take the form:

Note that the coupling may not be symmetric between the compartments since it is defined with respect to the surface area of each compartment. When the neuronal model is discretized and the biophysical mechanisms are specified for each compartment, the neuron looks strikingly similar to a complex integrated circuit (Fig. 18) and is ready to be tested in silico using neuron simulation software packages such as Neuron (http://www.neuron.yale.edu/neuron/) and Genesis (http://genesis-sim.org/).

Figure 18: A multi-compartment model of a neuron. The expanded region shows three compartments at a branch point where a single cable splits into two. Each compartment has membrane and synaptic conductances, as indicated by the equivalent electrical circuit, and the compartments are coupled together by

resistors. Although a single resistor symbol is drawn, note that gμ,μ' is not necessarily equal to gμ',μ .

4. Bibliography

Several good books have been used to put together these lecture notes, and most of the credits for the scientific accuracy and figures go to these books. In these brief lecture notes I tried to make a selection of topics that when discussed during a 4h lecture, have a certain consistency and give a feeling for the complexity with which neurons process their inputs within their dendrites. I left aside a lot of the biology since this course is aimed for physics students. This doesn’t mean that the biological aspects can be neglected; it simply means that there isn’t enough time to go in such detail and the reader is advised for a more rigorous treatment to refer to specialty books.

Books used to prepare lecture notes and slides:

1)Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems,

Peter Dayan and L.F. Abbot, MIT Press, Cambridge, Massachusetts, London, England, 2001.

2)Fundamental Neuroscience, Squire, Bloom, McConnell, Roberts, Spitzer, Zigmond, 2nd. ed., Academic Press, 2003.

3)Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Eugene M. Izhikevich, MIT Press, Cambridge, Massachusetts, London, England, 2007.

4)Principles of Neural Science, Eric R. Kandel, James H. Schwartz and Thomas M. Jessell, 4th ed., McGraw-Hill, 2000.

Popular Neuroscience books that I recommend (for e.g. bedtime reading or boring holidays):

5)In Search of Memory: The Emergence of a New Science of Mind, Eric R. Kandel, W. W. Norton & Company, 1st ed. 2007.

6)Rhythms of the Brain, Gyorgy Buzsaki, Oxford University Press, 1st ed. 2006.