Network analysis of the state space of discrete dynamical systems

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Dynamics and Complexity of Computrons

We investigate chaoticity and complexity of a binary general network automata of finite size with external input which we call a computron. As a generalization of cellular automata, computrons can have non-uniform cell rules, non-regular cell connectivity and an external input. We show that any finite-state machine can be represented as a computron and develop two novel set-theoretic concepts: (i) diversity space as a metric space that captures similarity of configurations on a given graph and (ii) basin complexity as a measure of complexity of partitions of the diversity space. We use these concepts to quantify chaoticity of computrons’ dynamics and the complexity of their basins of attraction. The theory is then extended into probabilistic machines where we define fuzzy basin partitioning of recurrent classes and introduce the concept of ergodic decomposition. A case study on 1D cyclic computron is provided with both deterministic and probabilistic versions.

Networks of recurrent events, a theory of records, and an application to finding causal signatures in seismicity

We propose a method to search for signs of causal structure in spatiotemporal data making minimal a priori assumptions about the underlying dynamics. To this end, we generalize the elementary concept of recurrence for a point process in time to recurrent events in space and time. An event is defined to be a recurrence of any previous event if it is closer to it in space than all the intervening events. As such, each sequence of recurrences for a given event is a record breaking process. This definition provides a strictly data driven technique to search for structure. Defining events to be nodes, and linking each event to its recurrences, generates a network of recurrent events. Significant deviations in statistical properties of that network compared to networks arising from (acausal) random processes allows one to infer attributes of the causal dynamics that generate observable correlations in the patterns. We derive analytically a number of properties for the network of recurrent events composed by a random process in space and time. We extend the theory of records to treat not only the variable where records happen, but also time as continuous. In this way, we construct a fully symmetric theory of records leading to a number of results. Those analytic results are compared in detail to the properties of a network synthesized from time series of epicenter locations for earthquakes in Southern California. Significant disparities from the ensemble of acausal networks that can be plausibly attributed to the causal structure of seismicity are as follows. (1) Invariance of network statistics with the time span of the events considered. (2) The appearance of a fundamental length scale for recurrences, independent of the time span of the catalog, which is consistent with observations of the "rupture length." (3) Hierarchy in the distances and times of subsequent recurrences. As expected, almost all of the statistical properties of a network constructed from a surrogate in which the original magnitudes and locations of earthquake epicenters are randomly "shuffled" are completely consistent with predictions from the acausal null model.

Chaotic gene regulatory networks can be robust against mutations and noise

Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as damage propagation, do not provide much information about robustness to mutations or noise in model gene regulatory networks.