The computational units of the brain: neuron and column models

The computational units of the brain: neuron and column models

century, innovative staining techniques of the nervous system by pi- onneers of histology, like Gerlach and Golgi, were used to support the reticular theory of cerebrospinal organization. The brain tissue was considered to be a continuous net of nerve fibers with holistic properties, being an exception to the cell theory. At the end of the 19 , Ramon y Cajal improved Golgi stainingtechniques and contributed to the opposite theory: the neuron doctrine. The reticular theory was also in contradiction with the localization of function in the brain like the Broca area dedicated to speech production and discovered in 1861. Sherrigton also supported the neuron theory and named synapse the connexion point between two neurons. The neuron is composed of a dendritic arbor on which presynaptic neurons make contact at dendrites, a cell body, also called soma and an axon. If inputs incoming to the cell body are sufficient, a spike is initiatiated at the axon hillock and propagates through the axon. These parts are illustrated on fig 2.1 for a generic neuron but many cell types with their specific morphology are found in the cortex. This all-or-none behaviour was used to design simplified models capturing the computational properties of the neuron, that is the way inputs are combined before deciding whether to spike or not. In a simple example of such artificial neurons, originally proposed by McCullough and Pitts [77], a weighted sum of the inputs is passed through a sigmoid transfer function. Having interesting computational properties, like any boolean function can be implemented by a network of such units, this arti- ficial neural network, sometimes with different transfer functions and additional learning dynamics on the weights, were a key element in the development of cybernetics and more specifically connectionism. Beside such formal approach, the understanding of the biophysical mechanisms responsible for spike genera- tion and propagation resulted in more realistic models of the neuron dynamics which will be presented in this chapter.

The modular theory of the brain and observations in histology led to the dis- covery of another set of computational unit at a mesoscale: the cortical columns. Cortical columns have been defined on anatomical ground with minicolumns of 50µm width containing around a hundred of cells being the result of cell mi- gration during development [78]. The macrocolumn is defined on a functional ground, as described in the introduction, but its anatomical subtrates are esti- mated of around 300µm for the visual cortex of the macaque monkey in [79]. A typical macrocolumn contains few thousands of cells and the detailed model of a cortical column of the rat somatosensory cortex in the Blue Brain project contains 10000 neurons of 200 possible types in a space of 500µm width and 1.5mm depth [80]. The column gathers cells coding for the same feature of the inputs so that a feature is reflected in the activity of a population of cells rather than in a single cell spike train. This redundancy in the vertical direc- tion of the cortex makes the code robust to perturbations of the dynamics like synaptic transmission failure or intrinsic fluctuations in cortical dynamics. First Depending on the animal and the area considered, the neuronal computa- tions can be understood at the single cell level or at the column level, it is thus necessary to analyze models of these computational units with tools from the theory of dynamical systems which will be presented in this chapter. Biolog- ically realistic models of the neuron have several variables (4 in the classical Hodgkin-Huxley model) often following a non-linear evolution equation making their analysis a difficult task. Reduced low dimensional models capture the es- sential features of the dynamics taking advantage of linearly related variables in the Fitz-Hugh Nagumo model or caricaturating the spike by an instantaneous reset after the membrane crosses a threshold. Models of a cortical column, with their huge state space, can also be reduced by considering the mean field ap- proximation of the network. In this chapter, after presenting the neuron and the cortical column, we give a short introduction to dynamical systems and then apply such methods to models of the computational units of the nervous system.