Input to output transfer in neurons

Computational modelling is playing an increasing role in neuroscience research by providing not only theoretical frameworks for describing the activity of the brain and the nervous system, but also by providing a set of tools and techniques for better understanding data obtained using various recording techniques. The focus of this thesis was on the latter - using computational modelling to assist with analyzing measurement results and the underlying mechanisms behind them. The first study described in this thesis is an example of the use of a computational model in the case of intracellular in vivo recordings. Intracellular recordings of neurons in vivo are becoming routine, yielding insights into the rich sub-threshold neural dynamics and the integration of information by neurons under realistic situations. In particular, these methods have been used to estimate the global excitatory and inhibitory synaptic conductances experienced by the soma. I first present a method to estimate the effective somatic excitatory and inhibitory conductances as well as their rate and event size from the intracellular in vivo recordings. The method was applied to intracellular recordings from primary motor cortex of awake behaving mice. Next, I studied how dendritic filtering leads to misestimation of the global excitatory and inhibitory conductances. Using analytical treatment of a simplified model and numerical simulations of a detailed compartmental model, I show how much both the mean, as well as the variation of the synaptic conductances are underestimated by the methods based on recordings at the soma. The influence of the synaptic distance from the soma on the estimation for both excitatory as well as inhibitory inputs for different realistic neuronal morphologies is discussed. The last study was an attempt to classify the synaptic location region based on the measurements of the excitatory postsynaptic potential at two different locations on the dendritic tree. The measurements were obtained from the in vitro intercellular recordings in slices of the somatosensory cortex of rats when exposed to glutamate uncaging stimulation. The models were used to train the classifier and to demonstrate the extent to which the automatic classification agrees with manual classification performed by the experimenter

Stability of neuronal networks with homeostatic regulation

Neurons are equipped with homeostatic mechanisms that counteract long-term perturbations of their average activity and thereby keep neurons in a healthy and information-rich operating regime. While homeostasis is believed to be crucial for neural function, a systematic analysis of homeostatic control has largely been lacking. The analysis presented here analyses the necessary conditions for stable homeostatic control. We consider networks of neurons with homeostasis and show that homeostatic control that is stable for single neurons, can destabilize activity in otherwise stable recurrent networks leading to strong non-abating oscillations in the activity. This instability can be prevented by slowing down the homeostatic control. The stronger the network recurrence, the slower the homeostasis has to be. Next, we consider how non-linearities in the neural activation function affect these constraints. Finally, we consider the case that homeostatic feedback is mediated via a cascade of multiple intermediate stages. Counter-intuitively, the addition of extra stages in the homeostatic control loop further destabilizes activity in single neurons and networks. Our theoretical framework for homeostasis thus reveals previously unconsidered constraints on homeostasis in biological networks, and identifies conditions that require the slow time-constants of homeostatic regulation observed experimentally

Parallel controllers do not lead to increased stability.

<p>The maximal recurrence is plotted against the time-constant of the second feedback loop. The <i>τ</i>₃ time-constant of the system with single feedback is indicated by the arrow. The system with the extra feedback loop (solid curve) is always less stable than the system with a single feedback loop (dashed line), even if the additional feedback is much slower than the original one. The control loop is shown in the inset.</p

Effects of heterogeneity and noise on homeostatic stability of a network.

<p>A) The stability as a function of heterogeneity in the neurons’ homeostatic time-constants. The time-constants <i>τ</i>₁, <i>τ</i>₂, <i>τ</i>₃ for each neuron were drawn from gamma-distributions with means 10, 50 and 100ms, respectively, and a CV given by the x-axis. The curve represents the mean maximal recurrence allowed to ensure a stable system. It decreases with heterogeneity. Error bars represent the standard deviation over 1000 trials. Simulation of 10 neurons, connected with a random, fixed weight matrix. B) Noise does not ameliorate instability. The fluctuations in the population firing rate, due to both noise and oscillations, are plotted as a function of the network recurrence. Without noise, fluctuations are only present when the recurrence exceeds the critical value (dashed curve). With noise, the fluctuations are already present in the stable regime and increase close to the transition point (solid curve). The homeostasis in the system amplifies noise compared to the non-homeostatic system (grey line). Homeostatic time-constants were 10, 50 and 100ms.</p

Networks with longer feedback cascade are less stable.

<p>A) The effect of adding a third filter to a two filter cascade. The stability is expressed as the maximum recurrence allowed in the network before it becomes unstable (transient oscillations allowed). The system with three filters is always less stable than the two filter system. The time-constants were set <i>τ</i>₁ = 10, <i>τ</i>₂ = 50 in the case of two filters, and <i>τ</i>₁ = 10, <i>τ</i>₂ = 50, <i>τ</i>₃ = <i>x</i>, as well as <i>τ</i>₁ = 10, <i>τ</i>₂ = <i>x</i>, <i>τ</i>₃ = 50 for the three filter case. B) Stability versus the number of filters for various filter cascades. As a function of filter number, time-constants were set linear 10, 20, 30, 40, … (dashed), constant with slow final integrator 10, 500, 500, …, 500, 5000 (dot-dashed), or exponential 10, 20, 40, … (solid) and 10, 30, 90, …ms (thick solid). The inset show the time-constants for a cascade with 10 filters for the various cases. Except for the last case, stability decreases with the number of filters.</p

Single neuron homeostasis.

<p>A) Schematic illustration of the homeostatic model. The input current is transformed through an input-output relation and a filter. The input-output curve is shifted by a filtered and integrated copy of the output firing rate, so that the average activity matches a preset goal value. <i>F1</i> (time-constant <i>τ</i>₁) denotes a filter describing the filtering between input and output of the neuron; <i>F2</i> (time-constant <i>τ</i>₂) is a filter between the output and the homeostatic controller. B) The response of the model for various settings of the homeostatic time-constants. The value of <i>τ</i>₁ was fixed to 10ms (thin lines), while <i>τ</i>₂ and <i>τ</i>₃ were varied. Center plot: the response of the neuron can either be stable (top left plot; white region), a damped oscillation (top right plot, gray region), or unstable (bottom right plot, striped region). The surrounding plots show the firing rate of the neuron and the threshold setting in response to a step stimulus.</p