Quantification of degeneracy in Hodgkin–Huxley neurons on Newman–Watts small world network

• Degeneracy can be applied to any neuronal network as a formal measure.
• The correlation analyses show degeneracy and complexity have positive correlation relationship.
• The degeneracy is a general topologic characteristic of neuronal networks, which could be applied quantitatively in neuroscience and connectomics.

Degeneracy is a fundamental source of biological robustness, complexity and evolvability in many biological systems. However, degeneracy is often confused with redundancy. Furthermore, the quantification of degeneracy has not been addressed for realistic neuronal networks. The objective of this paper is to characterize degeneracy in neuronal network models via quantitative mathematic measures. Firstly, we establish Hodgkin–Huxley neuronal networks with Newman–Watts small world network architectures. Secondly, in order to calculate the degeneracy, redundancy and complexity in the ensuing networks, we use information entropy to quantify the information a neuronal response carries about the stimulus – and mutual information to measure the contribution of each subset of the neuronal network. Finally, we analyze the interdependency of degeneracy, redundancy and complexity – and how these three measures depend upon network architectures. Our results suggest that degeneracy can be applied to any neuronal network as a formal measure, and degeneracy is distinct from redundancy. Qualitatively degeneracy and complexity are more highly correlated over different network architectures, in comparison to redundancy. Quantitatively, the relationship between both degeneracy and redundancy depends on network coupling strength: both degeneracy and redundancy increase with complexity for small coupling strengths; however, as coupling strength increases, redundancy decreases with complexity (in contrast to degeneracy, which is relatively invariant). These results suggest that the degeneracy is a general topologic characteristic of neuronal networks, which could be applied quantitatively in neuroscience and connectomics.