Philosophy

Epistemic Foundations of Symbolic Dynamics

The usefulness of symbolic dynamics rests on finding “good partitions” of the phase space, e.g. by construction of generating partitions which allow to approximate individual points in phase space by sustained measurements of coarse-graining devices. This is not possible for “misplaced” partitions where an intrinsic grain remains which makes certain states epistemically unaccessible. We have demonstrated the emergence of quantum-like properties such as complementary observables and contextual topologies (REFREF).

The Figure shows the possibility of complementary projectors in a chaotic dynamical system. As in algebraic quantum theory, two observables A, B are complementary if no eigenstate of A is eigenstate of B and vice versa. In statistical mechanics, an eigenstate of an observable A can be identified with a phase space domain R where A assumes a constant value making all individual states in R epistemically indistinguishable.

Epistemic Foundations of Symbolic Dynamics

Emergence of Mental States and Cognitive Processes

The concept of contextual emergence (REFREF) has been proposed as a non-reductive relation between different levels of description of physical and other systems where the lower level description comprises necessary but not sufficient conditions for the higher level description. These are supplied by contingent contexts obeying particular stability conditions. We have shown that Chalmers’ definition of “neural correlates of consciousness” (NCCs) can be complemented in terms of contextual emergence where the sufficient conditions are provided by contextually given “phenomenal families” partitioning the neural phase space (REF). Other examples for contextual emergence are syntactic language processing (REF), the evolutionary formation of categories (REF) or macrostates in neural networks (REF).

The Figure illustrates Descartes’ “organ metaphor of the mind” (left) mounted together with a brain synchronization map (right). Both portraits together make up the “vase versus faces” ambiguity (REF).

Emergence of Mental States and Cognitive Processes

Computational Neuroscience

Symbolic Dynamics of Neurophysiological Data

ERP data are given as large ensembles of short nonstationary (transient) and noisy time series. Symbolic dynamics of ERPs describes intertrial coherence of polarity deflections by running cylinder entropies and related measures. We have provided heuristics for symbolic encoding of ERP data, such as the median encoding (REF), the half-wave encoding (REFREFREF) and the stochastic resonance analysis (SRA) based on the findings of threshold stochastic resonance (REF). Especially the SRA allows to discriminate ERPs for conditions where voltage averaging fails (REFREFREFREF ).

The Figure shows a 3-symbol encoding of a noisy signal exhibiting stochastic resonance at the extrema of the signal.
Symbolic Dynamics of Neurophysiological Data

Emergence in Complex Neural Networks

A leaky integrator (LI) unit is the most simple model neuron described by an ordinary differential equation (REF). We have shown that at least two recurrently connected LI units may form nonlinear neural oscillators possessing limit cycles, which are known, e.g., in thalamo-cortical pathways (REF). We studied networks of coupled LI units in order to model global properties of the EEG (REFREF). Moreover, we are also investigating the issue of contextual emergence in neural networks (REF).

The Figure shows simulated EEG power spectra obtained from recurrent network of 20, 100, 200, 500, and 1000 LI units whose synaptic connections were randomly drawn such that 80% excitatory and 20% inhibitory synapses have been created. The spectra are computed for the oscillatory phase transition where super-cycles emerge in the network’s topology.

Emergence in Complex Neural Networks

Neural Field Theories

Very large networks of LI units can be described by a continuum approximation (REF). Starting from the LI equation the sum over the nodes connected with one unit has to be replaced by an integral transformation of a neural field quantity, where the continuous parameter now indicates the position of a unit in the network. Correspondingly, the synaptic weights turn into a kernel function. In addition, for large networks, the propagation velocity of neural activation has to be taken into account. We discuss the solvability and invertibility of neural field equations for general synaptic kernels (REFREFREF) and their applicability to computational psycholinguistics (REFREF) and cognitive science in general (REF).

The Figure should just illustrate the continuum limit starting from a discrete neural network and approaching a continuous neural tissue.

Neural Field Theories