Optimization principles of dendritic structure

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The Green's function formalism as a bridge between single- and multi-compartmental modeling

Neurons are spatially extended structures that receive and process inputs on their dendrites. It is generally accepted that neuronal computations arise from the active integration of synaptic inputs along a dendrite between the input location and the location of spike generation in the axon initial segment. However, many application such as simulations of brain networks use point-neurons—neurons without a morphological component—as computational units to keep the conceptual complexity and computational costs low. Inevitably, these applications thus omit a fundamental property of neuronal computation. In this work, we present an approach to model an artificial synapse that mimics dendritic processing without the need to explicitly simulate dendritic dynamics. The model synapse employs an analytic solution for the cable equation to compute the neuron's membrane potential following dendritic inputs. Green's function formalism is used to derive the closed version of the cable equation. We show that by using this synapse model, point-neurons can achieve results that were previously limited to the realms of multi-compartmental models. Moreover, a computational advantage is achieved when only a small number of simulated synapses impinge on a morphologically elaborate neuron. Opportunities and limitations are discusse

Using the Green's function to simplify and understand dendrites

Neurons are endowed with dendrites: tree-like structures that collect and transform inputs. These arborizations are believed to substantially enhance the computational repertoire of neurons. While it has long been known that dendrites are not iso-potential units, only in the last few decades it was shown experimentally that dendritic branches can transform local inputs in a non-linear fashion. This finding led to the subunit hypothesis, which states that within the dendritic tree, inputs arriving in one branch are transformed non-linearly and independently from what happens in other branches. Recent progress in experimental recording techniques shows that this localized dendritic integration contributes to shaping behavior. While it is generally accepted that the dendritic tree induces multiple subunits, many questions remain unanswered. For instance, it is not known how much separation there needs to be between different branches to be able to function as subunits. Consequently, there is no information on how many subunits can coexist along a dendritic arborization. It is also not known what the input-output relation of these subunits would be, or whether these subunits can be modified by input patterns. As a consequence, assessing the effects of dendrites on the workings of networks of neurons remains mere guesswork. During this work, we choose a theory-driven approach to advance our knowledge about dendrites. Theory can help us understand dendrites by deriving accurate, but conceptually simple models of dendrites that still capture their main computational effects. These models can then be analyzed and fully understood, which in turn teaches us how actual dendrites function computationally. Such simple models typically require less computer operations to simulate than highly detailed dendrite models. Hence, they may also increase the speed of network simulations that incorporate dendrites. The Green's function forms the basis for our theory driven approach. We first explored whether it could be used to reduce the cost of simulating dendrite models. One mathematically interesting finding in this regard is that, because this function is defined on a tree graph, the number of equations can be reduced drastically. Nevertheless, we were forced to conclude that reducing dendrites in this way does not yield new information about the subunit hypothesis. We then focused our attention on another way of decomposing the Green's function. We found that the dendrite model obtained in this way reveals much information on the dendritic subunits. In particular, we found that the occurrence of subunits is well predicted by the ratio of input over transfer impedance in dendrites. This allowed us to estimate the number of subunits that can coexist on dendritic trees. We also found that this ratio can be modified by other inputs, in particular shunting conductances, so that the number of subunits on a dendritic tree can be modified dynamically. We finally were able to show that, due to this dynamical increase of the number of subunits, individual branches that would otherwise respond to inputs as a single unit, could become sensitive to different stimulus features. We believe that this model can be implemented in such a way that it simulates dendrites in a highly efficient manner. Thus, after incorporation in standard neural network simulation software, it can substantially improve the accessibility of dendritic network simulations to modelers

Mathematical modelling in neurophysiology: Neuronal geometry in the construction of neuronal models

The underlying theme of this thesis is that neuronal morphology influences neuronal behaviour. Three distinct but related projects in the application of mathematical models to neurophysiology are presented. The first problem is an investigation into the source of the discrepancy between the observed conduction speed of the propagated action potential in the squid giant axon, and its value predicted on the basis of the Hodgkin-Huxley membrane model. It is shown that measurement error and biological variability cannot explain the discrepancy, nor can the use of a three-dimensional model to represent the squid giant axon. If the propagated action potential achieved the travelling wave speed in the experimental apparatus, as assumed implicitly by Hodgkin and Huxley, then it is suggested that the model of the membrane kinetics requires modification. The second problem involves the generalisation of Rall's equivalent cylinder to the equivalent cable. The equivalent cable is an unbranched structure with electrotonic length equal to the sum of the electrotonic lengths of the segments of the original branched structure, and an associated bijective mapping relating currents on the original branched structure to those on the cable. The equivalent cable is derived analytically and can be applied to any branched dendrite, unlike the Rall equivalent cylinder, which only exists for dendrites satisfying very restrictive morphological constraints. Furthermore, the bijective mapping generated in the construction of the equivalent cable can be used to investigate the role of dendritic morphology in shaping neuronal behaviour. Examples of equivalent cables are given for spinal inter neurons from the dorsal horn of the spinal cord. The third problem develops a new procedure to simulate neuronal morphology from a sample of neurons of the same type. It is conjectured that neurons may be simulated on the basis of the single assumption that they are composed of uniform dendritic sections with joint distribution of diameter and length that is independent of location in a dendritic tree. This assumption, in combination with the kernel density estimation technique, is used to construct samples of simulated interneurons from samples of real interneurons, and the procedure is successful in predicting features of the original samples that are not assumed by the construction process