Construction of an Experimentally-Based Neuronal Network Model to Explore the Function of the Inferi

This study is a synthesis between experimental work and theoretical modelling aiming to understand the mechanism of the sub-threshold oscillations (STOs) observed in the neurons of the inferior olive (IO) nucleus. The model was constructed in a bottom-up approach: it started from the construction of a detailed model of physiologically and morphologically reconstructed IO neurons, progressed to a model of electrically coupled passive neurons, advanced to a simpli ed model of two electrically coupled neurons with active properties, and completed with a large network model of the IO nucleus. The pro ciency from level to level was motivated by the requirement to simplify the model of individual IO neurons without loosing essential features. In the rst part (chapter 2), IO neurons were stained, their dendritic morphology was reconstructed, and detailed compartmental models were built. It was found that cable properties estimated from these compartmental models were signi cantly di erent than the cable properties measured experimentally. The modelling study demonstrated that the electrical coupling between IO neurons could qualitatively explain these di erences. In this reconciliation process, estimation of the number and locations of the gap junctions was provided. It was shown that at least some of the gap junctions are close to the cell bodies. In addition, it was estimated that, for each neuron, the number of coupled neighbors is at least four. The possibility that the uncoupling between IO neurons is implemented by shunting inhibition was also studied, and it was found that this mechanism implies a reduction of the input resistance of the uncoupled neurons, both when the shunt is onto gap junctions that are located between dendritic shafts or between complex spines (i.e, more distal). Previous studies showed that the low-threshold transient calcium (LTT) current is the major conductance involved in the generation of the STOs. In the second part of the work (chapter 3), the kinetics of the LTT were extracted using the whole cell patch clamp technique. As the cells of the IO are electrically coupled, the estimation of the kinetics based on voltage clamp iv data was generally not reliable. With the aid of a simpli ed model of two electrically coupled cells, each consisting of a soma endowed with LTT channels connected to a passive dendritic cable, it was demonstrated that with a careful choice of the voltage clamp paradigm, errors could be minimized. The basic idea was to choose voltage ranges in which the clamped cell and the neighbors were maximally and minimally activated, respectively. Moreover, it was shown that non-clamped data contains information on the structure of the network (such as the coupling coe cient). The model also showed that the dendrites may not be passive, and may include a voltage dependent conductance, which could act to uncouple the cells when they are depolarized. Returning to the experiments, these conclusions were applied to extract the kinetics of the LTT with maximal accuracy. The main results were that: (1) the dependence of steady-state activation on voltage is relatively steep; (2) the activation time constant (5 to 15 msec) is fast relative to the inactivation time constant ( 50 to 200 msec) and to the time scale of the STOs (5-10 Hz); (3) the LTT has a window conductance around the resting potential; and (4) the maximal amplitudes of the LTT current are very variable. This variability could be due, at least partially, to heterogeneity in the density of the LTT channels. In the last part (chapter 4), the single cell was modelled as a `morpho-less', point neuron (no dendrites), consisting of a leak and a LTT conductances. The LTT kinetics were taken from the results of chapter 3. The LTT activation was assumed to be instantaneous, such that the model was reduced to 2 coupled di erential equations, whereby phase plane techniques could be utilized. Four di erent behaviors to current clamp were found, depending on the ratio of leak and maximal LTT conductances: (i) stable, (ii) spontaneously oscillating,(iii) conditionally oscillating and (iv) conditionally bi-stable. Rhythmic activity could emerge from the electrical coupling of two quiescent cells of di erent types. It was shown that this rhythmic activity evolved from the mutual attraction of the two xed point into a range of voltages where the slope conductance(s), for one or both cells, is negative. In a large network model, where each cell was assigned with leak and LTT conductances chosen at random from normal probability v functions, the probability of the network to oscillate depended on the number of cells. In an all-to-all (ATA) connectivity, the probability rst decreased (to almost 0) and then increased to values close to 1, as the number of neurons increased. In a next-nearest-neighbor (NNN) connectivity, the probability either decreased to zero, remained constant, or increased to values smaller than 1. The low percentage of oscillating slices of the IO could thus be explained in a decrease in the number of coupled cells resulting from the slicing procedure, especially in a network with ATA connectivity. The behavior of a network model with the ATA connectivity was sensitive to current injection: oscillation could be formed, or abolished, by injection of current in a single cell of the network. In contrast, the behavior of a network with a NNN connectivity was resistant to current injection, as in the experiments: for example, in an oscillating network, oscillations were not abolished, and their frequency was almost una ected by current injection. It is concluded that: (1) the rhythmic activity in the IO could evolve from a network mechanism; (2) According to this model, the heterogeneity in channel densities between neurons plays a major role in the genesis of the STOs; and (3) The connectivity in the IO is intermediate, between NNN and ATA. vi