Estructura de los Centros de costos y tratamiento de las órdenes de producción.

Synapses are essential because they are the means by which neurons transmit signals from one to another. They mainly connect axons to dendrites, whereas there are many exceptions, e.g. dendro-dendritic synapses [105–108] and axo-axonic synapses [106, 109]. Depending on whether or not biochemical processes are involved, they can be classified into two fundamentally different types, chemical and electrical. Electrical synapses (gap junctions) are included in the sum-over-trips framework as boundary conditions, because they permit direct ion flows between coupled cells, and are thus simply modelled as purely passive resistors. No biochemical processes are involved, and thus signal transmission by gap junctions is metabolically inexpensive. It is observed in §4.4 that the gap junctional strength modulates signal ampli- tudes but has little effect on signal phases. The result is trivial because the gap junction plays the role of a resistor in the electrical circuit. The gap junctional location significantly modulates both amplitudes and phases of signals. However, the location of gap junction is in fact a geometric measurement of the dendritic

branches on which the gap junction is located, and the dendritic branches are reso- nant. Hence, computational results of the theoretical model could assist parameter estimation of gap junctions, given the knowledge of the dendritic geometric and electrial measurements; it is difficult to accurately measure the parameters of gap junctions in experiments due to their small sizes, but relatively easy to stimulate and record at somata.

Synaptic interactions

At a chemical synapse, the pre-synaptic neuron releases neurotransmitters (typically due to a spike) from synaptic vesicles into the synaptic cleft. The neurotransimitters can diffuse to the opposite side of the cleft, and if they bound to the corresponding ligand-gated channels of the post-synaptic cell, generation of post-synaptic currents could be triggered as these channels would change their conductances. At the same time, it is clear from Eq. (1.8) that the total membrane conductance is voltage dependent. Therefore, any post-synaptic current is also dependent on the temporal membrane potential at its location, which implies the interactions between synaptic inputs are inevitable. Such non-linear synaptic interactions on dendrites were dis- covered for a long time [64]. However, due to the model choice, Green’s functions found by our method always admit the additivity of multiple inputs (1.53), which can hardly be justified when multiple inputs are presented. Hence, Eq. (1.32) is only an idealised model of EPSC in the case of no shunting currents that would have varied the membrane conductance are presented. This model can be useful for experiments investigating single neuronsin vitro, though.

Considering shunting currents on an infinite cable, a more realistic model [110] for the post-synaptic current is

I0(x;t) =w(x;V)(E0−V), (5.8) where w(x;V) is membrane conductance at x, and E0 is the effective membrane reversal potential determined by all the ion channels. Since w(x;V) is voltage de- pendent, the input currentI0(x;t) becomes non-linear (inV). Although the system is no long a LTI system, the membrane potentials can still be written down in terms of a Green’s function: V(x;t) = Z t 0 Z ∞ −∞ G(x, y;t, s)I0(y;s)dyds+ Z ∞ −∞ G(x, y;t,0)V(y; 0)dy. (5.9) Notably the concept of Green’s function is here extended from a simple tool for

solving linear differential equations to a general object quatifying the relationship between input and output. It can be shown that the Green’s function G(x, y;t,0) is linear in space but non-linear in time, which leads to

G(x, y;t, s) =G(x−z1;t, tN−1)∗G(z1−z2;tN−1, tN−2)∗ · · · ∗G(z1−y;t1, s), (5.10) by the convolutional property (1.49), fors=t0 < t1 <· · ·< tN−1 < tN =t. If the time steps|tn−tn−1|, n∈ {1,2,3, . . . , N}are small enough, the non-linear Green’s functionG(zn−zn−1;tn, tn−1) is tiny, and can thus be approximated in terms of the linearG(zn−zn−1;tn−tn−1). Assuming this linear approximation works for all n and taking the limitN →+∞, Bressloff and Coombes [110] showed

G(x, y;t, s) =e−

Rt

sw(t

0_)dt0

G(x−y;t−s), (5.11) whereG(x−y;t−s) is the Green’s function for the model without shunting currents. Noticing the similarities between the proof in [110] and the deduction of the original sum-over-trips in [18], we should expect this result to be compatible with the sum- over-trips framework. Hence, synaptic interactions by shunting currents can be studied on dendritic trees by the same approach taken in this thesis.

To consider a typical neuronin vivo, that could have several thousand synapses, and constantly receive numerous synaptic inputs, background synaptic noise of this neuron has to be taken into account. Bressloff and Coombes [110] shows it is pos- sible to determine the effects of such noise on this neuronin vivo, by experimental investigation of the neuronin vitro. Green’s functions still play a role in the results, but more assumptions and techniques beyond the scope of this thesis are employed. A cost must be paid, if one attempts to understand the behaviours of single neurons in the context of a large network.

Synaptic plasticity

Another significant feature of synapses is synaptic plasticity, that is, the strength of a synapse can vary based on its activities. Synaptic plasticity is believed to be one of the most basic adaptation processes occuring in nervous systems, that ultimately enables learning behaviours of any creature with a nervous system [43]. Hebbian theory [111] offers the most famous explanation for synaptic plasticity, which is often summarised roughly as “cells that fire together wire together”. The idea is also widely employed in artifical neural networks, e.g. the Hopfield model [112].

Figure 5.1: Spike-timing dependent plasticity: the normalised change of synaptic strength as a function of the timing difference between the pre- and post-synaptic spikes, where wij is the synaptic strength between neuron i, j, ∆wij is its change, andti and tj are the spiking times of the two cells, respectively. Copied from [113]. Explicitly, they often employ the generalised Hebb’s rule,

∆wij =ηxixj, (5.12) wherexi andxj are the activities of neuroniand jrespectively, ∆wij is the change in the synaptic strength between them, and η is the learning rate. Although the generalised Hebb’s rule (5.12) is as simple as a bilinear form in the activities of the pre- and post-synaptic neurons, the Hebb’s rule commonly employed in biological neural networks, known asspike-timing dependent plasticity(STDP), is asymmetric and non-linear (see Fig. 5.1), which reveals the importance of temporal precedence in spikes. Notably these Hebb’s rules are mainly concerned with chemical synapses and the function of STDP could imply the casuality between spikes in pre- and post-synaptic neurons as the signal propagation is uni-directional. However, the plasticity of electrical synapses are often difficult to measure experimentally and had been poorly investigated until recently Turecek et al. [114] found a mechanism of coupling enhancement at the inferior olive electrical synapse.

Synaptic plasticity is not considered in this thesis at all. It mainly contributes to emergent properties of large neural networks, while the study of dendritic membrane potential dynamics focuses on single neurons or small neuronal circuits. They also live in different time scales; synaptic strengths are hardly changed after a few spikes.