La comprensión relacional de los problemas de adaptación

The PIR mechanism is a challenging system to broadly sweep and analyze, as was done within the escape-release framework discussed in previous chapters. The range of inhibition strengths for which PIR activity can exist is relatively narrow compared to the other mechanisms investigated. When this behavior does exist, there are always significant dead zones of system quiescence for the mechanism in the (gij, Iapp) bi-parametric state space. TWs remain the dominant stable rhythmic

pattern for most ranges and initial conditions for this system. PM rhythms require particular conditions to exist within this framework, and are nearly always a minor rhythm regime when compared to TW rhythms or static equilibrium. PIR networks jump quickly to final rhythm states due to both the required strong coupling and its inherent hard-locking nature. This makes PIR a viable candidate for CPG decision making in complex logic systems.

Results for symmetric near left-knee proximity of the nullclines indicate that PIR may induce transitions from TW-specific network outputs to PM rhythms. This results directly from changes in the duty cycle of the cells in the network as the move from smaller to larger values for affected coupling strengths. This relationship between duty cycle and rhythmic output is directly opposite that observed in both the escape and release cases, where TW typically remain dominate stable outputs with large duty cycle values. This could be a result of the hard-locking nature of PIR, and merits additional comparison to escape and release cases with definite hard locking characteristics. On the other end of the spectrum, when near right-knee proximity of the nullclines exists, transitions to PM behaviors are significantly reduced and only exist transiently, with TW rhythms occurring with much greater frequency. Within asymmetric PIR motifs, additional deviations from release-escape stereotypical behavior are observed, and most particularly for the clockwise-biased system. The general response of clockwise-biased increases in coupling strength, causing the emergence and or domination of a singular TW rhythm well above or below symmetry, holds true for only one of the two asymmetric synapse cases investigated. The other loses stability for all network output with decreasing clockwise anti-symmetry, becoming dominated entirely by quiescent output. This illustrates how much more dependent the PIR mechanism is on non-synaptic parameters than are the other mechanisms, with some cases where no magnitude of synaptic anti- symmetry can induce network outputs.

5.5 Discussion and applications

An examination of these further rhythmicity patterns, bifurcations, and the addition of additional asymmetry in several of the systems via coupling strength shifts or changes in fast-slow separation, aids in understanding these networks in a broader context in which manipulation of increasing numbers of parameters, plasticity, or residence within the framework of most of these patterns

occurs because of saddle-node bifurcations, often either homoclinic or heteroclinic. The unexpected regularity with which different phase-slipping behaviors can be observed in all three of the asymmetric motifs explored further lends itself to analysis of macro-scale rhythmic behaviors in which we may see periods of apparently stable patterns interspersed by fast rhythm switching to another apparently stable rhythm even without the need for external stimuli. These may present novel applications to experimental research of small local networks in which multi- stable rhythm production can be observed with the same network connectivity.

Further examination of changes in fast-slow separation within a system reiterate the increased dominance of oscillatory behavior with TW behaviors within the entire (gij, Iapp) bi-parametric

state space. This has been demonstrated directly both in this chapter and in the previous one, in which bifurcation detail across changing values of ε was performed more extensively both for symmetric and mono-biased networks, and is a direct result of the increased drive of the fast-cubic nullcline in drawing trajectories toward it relative to the slow sigmoidal nullcline. This leads to squarer limit cycle orbits and waveforms, as little or no clustering of cells can occur for very long at the knees of the cubic nullcline, even with near-knee proximity in either the release or escape case ranges of Iapp. Changes in increased duty cycle for escape cases was described specifically

here in the context of fast-slow separation as well, but are valid with increases in Iapp > 0.55 for all

network motifs explored, as well as within the framework of PIR.

In results described for exploration of PIR, there was a relatively narrow range of inhibitory coupling strengths for which bursting activity can be induced. When it does exist, it is generally a minority relative to the large zones of quiescence typically observed. TWs remain the dominant rhythmic output of the network for most ranges and initial conditions, with PM rhythms requiring highly restrictive conditions and typically coexisting with TW patterns and for an even more

restricted volume of IC space. PIR networks jump quickly to final rhythm states due to both the required strong coupling and its inherent hard-locking nature. Within asymmetric PIR motifs, additional deviations from release-escape stereotypical behavior are observed, most emphatically for the clockwise system, and underlines the increased dependence of the PIR mechanism on non- synaptic parameters.

6 TRANSITIONS IN 3-CELL MODULAR NETWORKING

Work in previous chapters has extensively explored the nature and behavior of local three-cell networks, with allusion to use of these not only in hypothesis generation for experimental studies, but also as a potential framework for using results from this work as 3-node building blocks for modular networking. Larger networks formed in this manner become more complicated, and visualization of dynamics observed in these systems becomes challenging within the framework of trace analysis and even using the phase-lag reduction employed thus far to reduce visualization to a two-dimensional system that could be displayed using the Poincare return maps extensively employed in this research. Some discussion is made here about ways in which higher-order networks could be readily analyzed and visualized, beginning with the simpler case of connection of a single additional node to one of the five key network motifs explored. This becomes more complicated with the addition of two cells, or the combination of two motifs, and an additional approach to doing this while continuing to use this methodology is described later in this chapter. Initial observations and results connecting 3-cell motifs into larger six-cell networks is described in the context of inhibitory connectivity, maintaining the reciprocally inhibitory HCO dynamics explored extensively thus far. Use of excitatory connections is not described here, but would be another method in which one network could be used to stimulate rhythm switching in the other, and is a recommended direction for future work building on this research. Finally, exploration of the addition of electrical between motifs is examined, with emphasis on transitions in rhythmogenesis and rhythm switching with increasing electrical connectivity. This approach is specifically used to describe a method for creating five-cell networks using strong electrical connection to effectively merge two cells together. This approach can also be used, and is described for a couple examples in the Appendix, to define even four-cell networks as coupled

three-cell motifs with two cells from each network being strongly electrically coupled. Broad transitions in rhythmicity are described, with the hope that future research can use some of these preliminary guiding principles in hypothesis generation and in furthering extensive bi-parametric exploration of these systems similarly to what has been done for the five key motifs described in this research, effectively building a growing library of bifurcation diagrams that can aid in continuing to bridge the gap between well-known small local network dynamics and behavioral outputs and rhythm generation in much larger interconnected networks.