One of the data sets we worked with in this case study is two hundred seconds of activity of a simulated neural network undergoing memory recall processes. This particular neural network simulation was of particular interest to us as it revealed an inherent behaviour which manifested itself in the spontaneous, i.e. without external stimuli, self-activation of stored memory patterns.
Seeking ways to grasp this behaviour we focused on the mutual relationships or interdependence that nodes have with each other across the network by computing the value of correlation of their activities. These values, calculated for each pair of the 81 neurons in the network, constructs a multidimensional structure which evolves, folding and unfolding in time. This abstract structure is placed in a space whose dimensions express the relationship between all possible node pairs (81! = 81*80*79*....*2*1).
In order to visualise this structure we searched for an operation which would transform a high dimensional object into a two dimensional figure. To this end, we devised another dynamical system, which would accomplish this specific task in an iterative process. This system is formed by 81 mutually interacting masses placed on a plane, one for each neuron. The magnitude of the force that each mass pair is subject to reflects the correlation value of that neuron pair: a higher correlation means greater attraction and therefore a smaller distance. A set of correlation values of the neural network activity would simultaneously cause all of the masses to move and search positions whose relative distances to all other masses correspond to that node's relationship to all other nodes. Similarity and interdependence are transposed into geometrical distance relationships. Eventually the dynamical system will result in an arrangement of the masses which reflects the best possible two dimensional approximation of the multi-dimensional structure, constructing a figure that folds and unfolds in time.
Mathematically, solutions of this operation, if any, are mostly non-unique: the task the dynamical system is set to take on is a hard problem. And when the system is pushed to the limits of its capabilities to interpolate between the two spaces, these difficulties become evident and the non-neutrality of the operation we are performing becomes clear. The dynamical system suddenly becomes material. It evolves from a problem solving, dimension reducing or simplifying operator and reveals itself as a form generating agent; it pushes back. Its distinct own behaviour becomes apparent.
In the end we find ourselves dealing with a more complex situation: two interacting and inextricably interwoven dynamical systems whose responsibilities in the result cannot be exactly separated. Clearly we formulated a transposition into a complexification, whose principal value lies less in the reduction function or in the calculation of an output, but in bringing to light qualities of such systems which are inherently incalculable.
We shed light on this situation by looking at it from different perspectives: we find multiple parametrisations for the forces and the figure's visual rendering. The result is a field of figures, artefacts whose mutual relationships construct a network through which incomputable qualities of the involved elements shimmer.