- Anatoly Buchin

# Invariant dynamics of neurons and networks

This post is inspired by the following __work__ of Eve Marder and Adam Taylor. I was always wondering how interesting things are tuned in biology. In particular how does the biophysics of single cells is adapted for the particular type of dynamics needed to serve computation. In general it is hard to determine what biological function exactly means, especially if you consider the behavior of just one part of the system such as single neuron in the nervous system.

Nevertheless if we forget for a while about real computations in the brain and concentrate on dynamics, it could still provide some insights. For example if we look into various combinations of channels in the membrane, their strength and kinetics, it seems that the resulting behavior of a neuron remains largely invariant when the cell operates in the circuit, see the picture below. On this picture are present the voltage traces of the stomatogastric neuron model, where on the bottom are maximal conductance parameters in the model, red shows the difference between the parameters. As you can see despite the variations the voltage traces are not that different. In this example the neural dynamics is preserved even when parameters are not exactly the same. In other words solutions of the dynamical system describing this neuron convergence towards large manifold, where dynamics is more-or-less the same in terms of voltage potential.

Indeed In my work related epilepsy I have discovered similar convergence of solutions, yet the level of the neural network model. As it was mentioned by Victor Jirsa 2014 in this __work____, when it comes to epilepsy the brain dynamics converges towards very narrow low-dimensional manifold. In my neural network model when I was looking for epileptic behavior I found similar convergence of the solutions as above. Percentages here are the amount of connection strength in the network. When the system is below the threshold, the network generates various oscillations or asynchronous state, where the dynamic manifold is rich and parameter-sensitive, black region on the curve. But once there is enough self-excitation, the seizure starts. This seizure dynamics (Ictal discharge) is very stable and stereotypical even if we vary the amount of synaptic connections, which makes it similar to the convergence of the solutions mentioned by Marder and Taylor. Typical seizure trace is depicted on the bottom.__

Yet when it comes to the network level, there is also particular amount of diversity. Beyond the threshold seizure seem to have different frequencies. But the behavior of the network looks very similar since all the cells participate in the form of population bursting. Thus we can see the convergence of the spiking patterns to the stereotypical bursting which is similar to convergence of solutions towards the same dynamical manifold in single neurons from somatogastric network.