Theoretical neuroscience: from long wavelength cortical patterning to spatial navigation

The science of brain function has a long and vibrant history. Recent technological developments have dramatically improved and facilitated data acquisition from a variety of methodologies to monitor brain activity, ranging from electroencephalography to opto-genetics. This highlights a need for concomitant theories of brain function. Such theories can act as a bridge between descriptions of the brain pertaining to data at different levels, from molecular to behavioural, using methods of mathematics, physics, and computer science. The models presented in this thesis do not incorporate all the biophysical, anatomical and physiological data collected to date. Rather, the focus is on simplified models that contain sufficient detail to explain the essence of the phenomena considered. Moreover, they are constructed to allow the application of analytical mathematical tools to explore their behaviour. In particular, this thesis proposes parsimonious neural models that aim to explain the mechanism by which humans and animals can navigate using spatial memory. The material presented ranges over a number of levels of description, and utilises a variety of mathematical techniques. A common theme throughout is the use of ideas from nonlinear dynamical systems to gain insight into neural mechanisms, ranging from activity patterns of cells underlying navigation, to the derivation of temporal difference reinforcement learning algorithms to solve reward based problems. This work presents three main contributions. Firstly, it analytically determines which model parameters contribute to the observed difference in wavelength scale of the formed activity patterns in computational models for grid cells. Moreover, this thesis explores extensions to these models in order to find a neural mechanism that could account for the difference in wavelength scale. It is shown, after analysing the linear stability of spatially homogeneous steady states to spatio-temporal perturbations, that the addition of axo-dendritic connections provides a mechanism for the difference in wavelength scale. Secondly, based on recent research, this work proposes a different type of model, a network of spiking neurons, to uncover the mechanisms, related to rebound spiking, for variation in scale of grid cell firing fields. Travelling waves are observed on computer simulations of this model. The analytical construction of such waves is accomplished using techniques from the field of non-smooth dynamical systems. Moreover, the dispersion curve, that determines how wave speed varies as a function of the period, is constructed. Such dispersion curve exhibits a wide range of long wavelength solutions. In order to exhibit how the variation of parameters affects the maximum allowed period, a wave stability analysis is developed. This work entails and broadens the use of non-standard analysis techniques. The final part of the thesis makes a direct link to experiments, combining reinforcement learning theory and computer simulations to shed light on the neurocomputational mechanisms underlying behaviour of rats in a variation of the Morris watermaze experiment. Particularly, the simulation employs a continuous time actor-critic framework, in which the actor and critic are represented as firing rate neural networks. The ability of the artificial rats to learn and reach the different goal locations is measured under different variations of the model

Theoretical neuroscience: from long wavelength cortical patterning to spatial navigation

The science of brain function has a long and vibrant history. Recent technological developments have dramatically improved and facilitated data acquisition from a variety of methodologies to monitor brain activity, ranging from electroencephalography to opto-genetics. This highlights a need for concomitant theories of brain function. Such theories can act as a bridge between descriptions of the brain pertaining to data at different levels, from molecular to behavioural, using methods of mathematics, physics, and computer science. The models presented in this thesis do not incorporate all the biophysical, anatomical and physiological data collected to date. Rather, the focus is on simplified models that contain sufficient detail to explain the essence of the phenomena considered. Moreover, they are constructed to allow the application of analytical mathematical tools to explore their behaviour. In particular, this thesis proposes parsimonious neural models that aim to explain the mechanism by which humans and animals can navigate using spatial memory. The material presented ranges over a number of levels of description, and utilises a variety of mathematical techniques. A common theme throughout is the use of ideas from nonlinear dynamical systems to gain insight into neural mechanisms, ranging from activity patterns of cells underlying navigation, to the derivation of temporal difference reinforcement learning algorithms to solve reward based problems. This work presents three main contributions. Firstly, it analytically determines which model parameters contribute to the observed difference in wavelength scale of the formed activity patterns in computational models for grid cells. Moreover, this thesis explores extensions to these models in order to find a neural mechanism that could account for the difference in wavelength scale. It is shown, after analysing the linear stability of spatially homogeneous steady states to spatio-temporal perturbations, that the addition of axo-dendritic connections provides a mechanism for the difference in wavelength scale. Secondly, based on recent research, this work proposes a different type of model, a network of spiking neurons, to uncover the mechanisms, related to rebound spiking, for variation in scale of grid cell firing fields. Travelling waves are observed on computer simulations of this model. The analytical construction of such waves is accomplished using techniques from the field of non-smooth dynamical systems. Moreover, the dispersion curve, that determines how wave speed varies as a function of the period, is constructed. Such dispersion curve exhibits a wide range of long wavelength solutions. In order to exhibit how the variation of parameters affects the maximum allowed period, a wave stability analysis is developed. This work entails and broadens the use of non-standard analysis techniques. The final part of the thesis makes a direct link to experiments, combining reinforcement learning theory and computer simulations to shed light on the neurocomputational mechanisms underlying behaviour of rats in a variation of the Morris watermaze experiment. Particularly, the simulation employs a continuous time actor-critic framework, in which the actor and critic are represented as firing rate neural networks. The ability of the artificial rats to learn and reach the different goal locations is measured under different variations of the model

An analysis of waves underlying grid cell firing in the medial enthorinal cortex

Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an I_h current in response to hyperpolarising synaptic input. A computational modelling study by Hasselmo [2013 Neuronal rebound spiking, resonance frequency and theta cycle skipping may contribute to grid cell firing in medial entorhinal cortex. Phil. Trans. R. Soc. B 369: 20120523] showed that an inhibitory network of such cells can support periodic travelling waves with a period that is controlled by the dynamics of the I_h current. Hasselmo has suggested that these waves can underlie the generation of grid cells, and that the known difference in I_h resonance frequency along the dorsal to ventral axis can explain the observed size and spacing between grid cell firing fields. Here we develop a biophysical spiking model within a framework that allows for analytical tractability. We combine the simplicity of integrate-and-fire neurons with a piecewise linear caricature of the gating dynamics for HCN channels to develop a spiking neural field model of MEC. Using techniques primarily drawn from the field of nonsmooth dynamical systems we show how to construct periodic travelling waves, and in particular the dispersion curve that determines how wave speed varies as a function of period. This exhibits a wide range of long wavelength solutions, reinforcing the idea that rebound spiking is a candidate mechanism for generating grid cell firing patterns. Importantly we develop a wave stability analysis to show how the maximum allowed period is controlled by the dynamical properties of the I_h current. Our theoretical work is validated by numerical simulations of the spiking model in both one and two dimensions

Biophysical Modeling of Actin-Mediated Structural Plasticity Reveals Mechanical Adaptation in Dendritic Spines.

Synaptic plasticity is important for learning and memory formation; it describes the strengthening or weakening of connections between synapses. The postsynaptic part of excitatory synapses resides in dendritic spines, which are small protrusions on the dendrites. One of the key features of synaptic plasticity is its correlation with the size of these spines. A long-lasting synaptic strength increase [long-term potentiation (LTP)] is only possible through the reconfiguration of the actin spine cytoskeleton. Here, we develop an experimentally informed three-dimensional computational model in a moving boundary framework to investigate this reconfiguration. Our model describes the reactions between actin and actin-binding proteins leading to the cytoskeleton remodeling and their effect on the spine membrane shape to examine the spine enlargement upon LTP. Moreover, we find that the incorporation of perisynaptic elements enhances spine enlargement upon LTP, exhibiting the importance of accounting for these elements when studying structural LTP. Our model shows adaptation to repeated stimuli resulting from the interactions between spine proteins and mechanical forces

Exploring new roles for actin upon LTP induction in dendritic spines

Dendritic spines, small protrusions of the dendrites, enlarge upon LTP induction, linking morphological and functional properties. Although the role of actin in spine enlargement has been well studied, little is known about its relationship with mechanical membrane properties, such as membrane tension, which is involved in many cell processes, like exocytosis. Here, we use a 3D model of the dendritic spine to investigate how polymerization of actin filaments can effectively elevate the membrane tension to trigger exocytosis in a domain close to the tip of the spine. Moreover, we show that the same pool of actin promotes full membrane fusion after exocytosis and spine stabilization

Biophysical 3D modeling of actin-mediated structural plasticity

<p>Simulation of dendritic spine enlargement upon LTP stimulation using PDEs for actin, arp2/3, and cofilin dynamics in a moving boundary framework. </p> <p>Run Simulation_main.m</p> <p>For the remeshing function (remeshing.m) see <a href="https://github.com/christopherhelf/isotropicremeshing/blob/master/README.md">https://github.com/christopherhelf/isotropicremeshing/blob/master/README.md</a> and the instructions therein.</p>This work was supported by the NIH Grant Number 1RF1DA055668-01 and by an Air Force Office of Scientific Research Grant FA9550-18-1-0051 to P.R

Reproducing asymmetrical spine shape fluctuations in a model of actin dynamics predicts self-organized criticality

Dendritic spines change their size and shape spontaneously, but the function of this remains unclear. Here, we address this in a biophysical model of spine fluctuations, which reproduces experimentally measured spine fluctuations. For this, we characterize size- and shape fluctuations from confocal microscopy image sequences using autoregressive models and a new set of shape descriptors derived from circular statistics. Using the biophysical model, we extrapolate into longer temporal intervals and find the presence of 1/f noise. When investigating its origins, the model predicts that the actin dynamics underlying shape fluctuations self-organizes into a critical state, which creates a fine balance between static actin filaments and free monomers. In a comparison against a non-critical model, we show that this state facilitates spine enlargement, which happens after LTP induction. Thus, ongoing spine shape fluctuations might be necessary to react quickly to plasticity events