Automated Parameter Search in Small Network Central Pattern Generators
Automated parameter search in small network central pattern generators (CPGs) involves the use of any methods other than manual (i.e., hand-tuning) to generate or tune sets of parameters that result in physiologically realistic neuronal models of the CPGs. Such methods include “brute-force” explorations of predefined parameter spaces, as well as various heuristics (e.g., multi-objective evolutionary algorithms) used to arrive at a single or more of viable model parameter combinations.
Central pattern generators (CPGs) are neural networks that produce rhythmically patterned outputs, without relying on any sensory feedback (Hooper 2001). CPGs drive such critical rhythmic activity as breathing, chewing, swimming, walking, heartbeat control, etc. CPGs have been shown to produce rhythmic outputs akin to normal rhythmic activity patterns, even in isolation from other parts of the nervous system, which makes them popular physiological models. Furthermore, due to their relative simplicity, especially in such model organisms as lobsters, crabs, or leeches, CPGs have also become quite widespread in computational modeling studies of cellular and synaptic properties of individual neurons and small neural networks.
While hand-tuning has been traditionally used in the process of creating CPG neuronal models (e.g., Soto-Treviño et al. 2005), in light of recent advances in the computational capabilities of modern computing systems, which now facilitate the use of more complex neuronal models (i.e., in terms of the number of compartments or free parameters), and allow for the exploration of unprecedentedly large parameter search spaces, this approach has become virtually obsolete. Therefore, automated methods for model parameter search have been lately gaining much attention.
There are basically two approaches to the problem of searching for optimal (i.e., physiologically realistic) sets of parameter values for models of small network central pattern generators: (1) “brute-force” explorations of predefined parameter spaces and (2) explorations utilizing heuristic optimization approaches, such as multi-objective evolutionary algorithms (MOEAs).
“Brute-Force” Explorations of Predefined Parameter Spaces
In the case of “brute-force” explorations of predefined parameter spaces, the study usually starts with a hand-tuned model of the CPG, which serves as the “center” for the parameter search space that is created around it. Then, physiologically realistic ranges for the model parameters (e.g., maximal conductances of membrane and synaptic currents) are chosen, along with the granularity for each of the parameters. The granularity determines how many possible values each of the parameters can assume and does not have to be the same for all the parameters, as some of them will exhibit different sensitivities to changes in their values. In some cases, the first step may be omitted and only the parameters, along with their ranges and granularities, are determined.
After such a grid-based parameter search space has been constructed, all of the possible combinations of the parameter values are simulated and tested for their physiological adequacy (possibly under multiple simulation scenarios, such as spontaneous activity, response to current injections, removal of neuromodulation, etc.). Only those models that match the behavior of the biological CPG, which is determined by means of one or more quantitative or qualitative measures of the CPG’s characteristics (e.g., spike height, inter-spiking interval, burst duration, period, preservation of the phase of the rhythm, etc.), are retained for further analyses. However, the rejected models are also sometimes subjected to examination in order to determine what makes “bad” models unacceptable.
Explorations Utilizing Heuristic Optimization Approaches
In the case of explorations utilizing various heuristic approaches, such as multi-objective evolutionary algorithms, the study usually starts with the determination of the model parameters, along with their ranges and granularities, similarly to the “brute-force” approach, but often on a much larger scale. In other words, while the range in the “brute-force” approach may reflect a 3- to 4-fold variation in the parameter values, and the granularity may allow for five to ten possible values, incorporating up to 20-fold variation with hundreds of possible values for each parameter is not unheard of in a heuristic approach. Since this approach is not tasked with simulation and analysis of all of the possible combinations of values in such created parameter search space, it remains computationally feasible.
Another critical step in this approach is the definition of one or more measures of the given CPG’s characteristics that will be used to determine physiological adequacy of the models. While such measures can be virtually identical to the ones used in the “brute-force” approach, the difference lies in the fact that they are being used during the process of model generation itself, rather than at the end to filter out the unwanted models. These measures, in essence, become the fitness functions utilized in the process of optimizing model parameter values to drive it toward generating as many as possible “good” models which match the biological system, while limiting the number of “bad” solutions.
After the model parameters and the corresponding ranges and granularities have been determined, and the appropriate fitness functions (possibly multiple, even conflicting) have been defined, the iterative process of optimization of the model parameter values begins and ultimately yields a collection of physiologically realistic CPG models.
Most of the hitherto applications of the automated parameter searches in small network central pattern generators have been performed in relatively simple invertebrate CPGs. For example, Doloc-Mihu and Calabrese (2011) utilized the “brute-force” approach to construct a large (on the order of terabytes) database of conductance-based models of the half-center oscillator from the leech heartbeat central pattern generator to determine how neuronal parameters influence the network activity. Using the same approach, Günay and Prinz (2010) utilized a large (20,250,000) database of models of the lobster pyloric network to study calcium sensors for network homeostasis. Smolinski, Prinz et al., used both the “brute-force” approach and multi-objective evolutionary algorithms to study the cellular and synaptic properties of the AB/PD (anterior burster/pyloric dilator) pacemaker kernel in the lobster pyloric network (2006, 2009), as well as the conductance correlations involved in the recovery of bursting after neuromodulator deprivation (Shim et al. 2012; Malik et al. 2013).
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