No two neurons are alike: degeneracy in neurons and neural circuits Open Access

Tian, Kun (Fall 2019)

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Neurons and neural circuits need to balance flexibility with stability to constantly adapt to the changing environment while maintaining stable outputs. One of the candidate mechanisms is degeneracy, which refers to the phenomenon that different combinations of a neuron or neural circuit’s parameters (i.e. degenerate solutions) can give rise to similar neural activity. Degeneracy has been widely observed in invertebrate and vertebrate neural systems, yet still little is known about relationships among the degenerate solutions and how they relate to the function of neurons and neural circuits. Combining experimental data with computational modeling, we explored the questions above in two neural systems: the pyloric circuit in the crab Cancer borealis and the sympathetic postganglionic neurons (SPNs) in mice.

The pyloric circuit generates a stereotypical rhythm, and multiple combinations of its cellular and synaptic parameters can produce similar pyloric rhythm. To explore the questions above, we measured the linear structures of the degenerate solutions, and found that reducing the variability of pyloric rhythm features, but not the number of parameters, led to increased strength of linear structures of the degenerate solutions.

SPNs, the last common motor output of the sympathetic nervous system, pass converged motor commands from the spinal cord to downstream muscles and visceral organs. SPNs located in the thoracic region (tSPNs) innervate vasculature, and a RNA- Seq study identified two types of vasculature-innervating tSPNs, NA2 and NA3, which differ in their cell size, but little is known about whether they differ in excitability and ability to integrate synaptic inputs. We built the first physiologically-realistic model of tSPNs in mice based on experimental data, and then built a database of tSPN model versions that match the cell size of NA2- and NA3-type tSPNs. We found that, compared to NA2-type tSPNs, NA3-type have lower densities of hyperpolarizing currents and higher input resistance, making them more excitable with greater ability to integrate synaptic inputs.

Together, these insights from examining a collection of degenerate solutions instead of a single one will help us better understand how neurons and neural circuits employ degenerate solutions to maintain stable outputs against perturbations and injuries. 

Table of Contents

1. General Introduction 1

1.1 Structures of degenerate parameter sets 2

1.1.1 Ion channel correlations and sloppiness 4

1.1.2 From linear to nonlinear structures 7

1.2 Degenerate solutions and homeostatic plasticity mechanisms 10

1.2.1 Single neuron level 11

1.2.2 Neural circuit level 13

1.3 Ensemble modeling 17

2. Degenerate solutions in the pyloric circuit of the crab Cancer borealis 21

2.1 Introduction 21

2.1.1 Background 21

2.1.2 Overview of the pyloric circuit in the crab Cancer borealis 23

2.2 Methods 27

2.2.1 Pyloric circuit model 27

2.2.2 Multi-objective evolutionary algorithm (MOEA) 29

2.2.3 Data analysis 31

2.2.4 Implementation and code accessibility 32

2.3 Results 34

2.3.1 Computational approach 34

2.3.2 Cellular and synaptic parameters before and after IMI removal 36

2.3.3 Ion channel correlations before and after IMI removal 40

2.3.4 Ion channel correlations before and after reducing the variability of features 43

2.3.5 Principal component analysis 45

2.4 Discussion 46

2.4.1 Summary of results 46

2.4.2 How constraints shape structures of degenerate solutions 47

2.4.3 Formation of ion channel correlations 49

2.4.4 Biological insights: Ion channel correlations at the neural circuit level 53

2.4.5 Strength of our computational approach 54

2.4.6 Limitations in our computational approach 55

2.4.7 Future directions: A theoretical framework of degeneracy 56

2.5 Acknowledgments 57

3. Degenerate solutions in tSPNs in mice 58

3.1 Introduction 58

3.1.1 Background 58

3.1.2 Overview of sympathetic postganglionic neurons in mice 60

3.2 Methods 61

3.2.1 Single neuron model of tSPNs 61

3.2.2 Synaptic inputs 63

3.2.3 Brute force search 64

3.2.4 Data analysis 65

3.2.5 Implementation and code accessibility 66

3.3 Results 67

3.3.1 Cellular mechanisms underlying firing properties of tSPNs when hyperpolarized 67

3.3.2 Cellular mechanisms underlying firing properties of tSPNs when depolarized 68

3.3.3 Model neuron is capable of integrating synaptic inputs 73

3.3.4 Two types of vasculature-innervating tSPNs 73

3.3.5 tSPNs with identical f-I curve differ in their ability to integrate synaptic inputs 78

3.4 Discussion 81

3.4.1 Summary of results 81

3.4.2 The influence of cell size on neuronal excitability 82

3.4.3 Firing mode: integrator vs. coincidence detector 83

3.4.4 Degenerate solutions and homeostatic regulation after SCI 84

3.4.5 Future Direction: Elucidating intrinsic homeostatic plasticity mechanisms in tSPNs after SCI 85

3.4.6 Future direction: Determining factors that influence synaptic integration in tSPNs 86

3.5 Acknowledgements 87

4. General Discussion 88

4.1 Homeostatic plasticity: tuners vs. targets 88

4.2 Homeostatic plasticity mechanisms: activity-dependent vs. activity- independent 92

4.3 Homeostatic plasticity and mathematical theory 94

4.4 Future directions on degeneracy in neurons and neural circuits 95

4.4.1 Defining measures of degeneracy 96

4.4.2 Degeneracy and robustness against injuries 97

4.5 Final thoughts 98

5. Appendix 101

5.1 Equations for pyloric circuit model 101

5.2 Equations for single neuron model of tSPNs 104

5.3 Supplemental data: Distribution of each parameter before and after IMI removal 107

5.4 Supplemental data: Ion channel correlations in tSPNs 114

6. References 115 

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