Work Package*: *WP2

Programme*: *P5

Deliverable*: *Deliverable 5.1: “Mathematical framework for spiking neural networks applied to cultures and robot control”

Deliverable due date: month 30

This document reports on the progress of work on Deliverable 5.1, regarding the development and dissemination of techniques from computational neuroscience to members of the experimental team at UNOTT developing work on cultures and broader mathematical modelling techniques to the team at UMinho working on robot control.

Cultures

Training in dynamical systems and the use of the Brian Python module (for spiking neural networks) was provided by ER1 (Sid Visser) to members of the UNOTT experimental team in a series of blackboard tutorials together with bespoke code in Python and Matlab (for mathematical models of spike-dependent plasticity that could be used to solve the *distal reward problem* to be used as a backdrop to understand how in principle cultures could be trained to solve computational problems).

Many long-term processes of developing neural networks are generally underemphasised in computational neuroscience. Rather than studying how the networks came into existence in the first place (a long-term process), the neural networks are often considered on various kinds of random graphs. In *in vivo* or *ex vivo* situations, the complicated network structure is unknown and cannot be reconstructed with current imaging techniques. This lack of knowledge, to some extent, legitimates a random network.

In primary cell cultures, the situation is quite the opposite. The neurons, namely, lose all their connections after being taken from a donor, such that the culture starts off as an unconnected network. Understanding how the neural networks evolve, due to both the physical growth of the neurons and their changes in electrophysiological activity, provides essential insights in the way brains develop and learn.

As a starting point, we have identified three different physiological mechanisms that play an important role in the development of cultured neurons: First of all, we have considered the physical growth of neurons in the network and the way they find and connect with other neurons. Thereafter, we have reviewed various homeostatic mechanisms inside a neuron that change the cell’s behaviour during its life-time. Finally, we have investigated how the biophysical processes, that underlie distal learning, impact network formation when a neural culture is exposed to and interacting with a virtual environment.

Network development is a well-studied topic, but the majority of computational models in this field are too detailed and irrelevant for the development seen in cultures, since cultures are two-dimensional structures. Therefore, we have development a much simpler and computationally efficient model that describes the growth of a neuron in terms of a cellular automaton. In the absence of neural activity, the neuron’s growth is described a continuous time Markov process on a two dimensional grid. When two neurons are touching or overlapping in physical space, a synaptic connection can be formed between the neurons.

Further to that, we have explored how learning takes places in a synthetic neural network. In particular, we have implemented the hypothesised mechanisms underlying distal reward learning into an *in silico* network. Next, we have designated some cells in the network as ‘sensory’ neurons and others as ‘motor’. This has enabled us to explore how a neural network can interact with a virtual environment and how certain (externally imposed) objectives can be achieved by the network as a consequence of its learning ability.

Robot control

On the secondment of ESR Weronika Wojtak from UMinho to UNOTT we developed a new mathematical model for localised bump attractors – which form the heart of current dynamic neural field paradigms for robot action at UMinho – to further encode for bump *amplitude* as well as position. This gives a new and important extra degree of freedom. The mathematical analysis of the system in one and two spatial dimensions has been completed and a joint paper is in preparation covering this and the application to robot control (based on the report “Neural field model supporting a continuum of bump amplitudes” May 2015, prepared by Weronika Wojtak).

Here is it possible to see a Computer simulation of the growth of three neurons in a culture. Each neuron is shown in a different colour and, as time passes, the neurons grow and explore their environment. In some cases, the neurons will retract their processes and extend in a different direction again. All these phenomena are also observed in real cultures of developing neurons.

Contributors: Sid Visser, Weronika Wojtak, Alessandro Barardi, Nitzan Herzog, Alban Levy, Steve Coombes, Noah Russell (UNOTT, UPC, UMinho)