Mathematic modelling of the enteric nervous network. 5. Excitation propagation in a planar neural network

Unique Neural Circuit Connectivity of Mouse Proximal, Middle, and Distal Colon Defines Regional Colonic Motor Patterns

BACKGROUND & AIMS

Colonic motor patterns have been described by a number of different groups, but the neural connectivity and ganglion architecture supporting patterned motor activity have not been elucidated. Our goals were to describe quantitatively, by region, the structural architecture of the mouse enteric nervous system and use functional calcium imaging, pharmacology, and electrical stimulation to show regional underpinnings of different motor patterns.

METHODS

Excised colon segments from mice expressing the calcium indicator GCaMP6f or GCaMP6s were used to examine spontaneous and evoked (pharmacologic or electrical) changes in GCaMP-mediated fluorescence and coupled with assessment of colonic motor activity, immunohistochemistry, and confocal imaging. Three-dimensional image reconstruction and statistical methods were used to describe quantitatively mouse colon myenteric ganglion structure, neural and vascular network patterning, and neural connectivity.

RESULTS

In intact colon, regionally specific myenteric ganglion size, architecture, and neural circuit connectivity patterns along with neurotransmitter-receptor expression underlie colonic motor patterns that define functional differences along the colon. Region-specific effects on spontaneous, evoked, and chemically induced neural activity contribute to regional motor patterns, as does intraganglionic functional connectivity. We provide direct evidence of neural circuit structural and functional regional differences that have only been inferred in previous investigations. We include regional comparisons between quantitative measures in mouse and human colon that represent an important advance in showing the usefulness and relevance of the mouse system for translation to the human colon.

CONCLUSIONS

There are several neural mechanisms dependent on myenteric ganglion architecture and functional connectivity that underlie neurogenic control of patterned motor function in the mouse colon.

Mathematical modelling of enteric neural motor patterns

1. The enteric nervous system modulates intestinal behaviours, such as motor patterns and secretion. Although much is known about different types of neurons and simple reflexes in the intestine, it remains unclear how complex behaviours are generated. 2. Mathematical modelling is an important tool for assisting the understanding of how the neurons and reflexes can be pieced together to generate intestinal behaviours. 3. Models have identified a functional role for slow excitatory post-synaptic potentials (EPSPs) by distinguishing between fast and slow EPSPs in the ascending excitation reflex. These models also discovered coordinated firing of similarly located neurons as emergent properties of feed-forward networks of interneurons in the intestine. A model of the recurrent network of intrinsic sensory neurons identified important control mechanisms to prevent uncontrolled firing due to positive feedback and that the interaction between these control mechanisms and slow EPSPs is necessary for the networks to encode ongoing sensory stimuli. This model also showed that such networks may mediate migrating motor complexes. 4. A network model of vasoactive intestinal peptide neurons in the submucosal plexus found this relatively sparse recurrent network could produce uncontrolled firing under conditions that appear to be related to cholera toxin-induced hypersecretion. 5. Abstract modelling of the intestinal fed-state motor patterns has identified how stationary contractions can arise from a polarized network. 6. These models have also helped predict and/or explained pharmacological evidence for two rhythm generators and the requirement of feedback from contractions in the circular muscle.

Perspectives on Mathematical Modeling for Receptor-Mediated Processes

Mathematical modeling is a potent in silico tool that can help investigate, interpret, and predict the behavior of biological systems. The first step is to develop a working hypothesis of the biology. Then by "translating" the biological phenomena into equations, models can harness the power of mathematical analysis techniques to explore the dynamics and interactions of the biological components. Models can be used together with traditional experimental models to help design new experiments, test hypotheses, identify mechanisms, and predict outcomes. This article reviews the process of building, calibrating, and using mathematical models in the context of the kinetics of receptor and signal transduction biology. An example model related to the androgen receptor-mediated regulation of the prostate is presented to illustrate the steps in the modeling process and to highlight the potential for mathematical modeling in this area.

Dendritic integration in a recurrent network

Classical nonlinear cable theory is appropriate for the unmyelinated axonal membrane because voltage-dependent ion channels are densly distributed, but dendrites with a sparse density distribution of voltage-dependent ion channels show "weakly" excitable membrane properties. Therefore, a model for "weakly" active dendrites is presented by introducing voltage-dependent ion channels at discrete locations along the dendritic cable. This provides an alternative representation for the investigation of regenerative potentials in dendrites in order to explore how active dendrites influence synaptic integration. As an example, we consider a two-neuron recurrent network of biophysically distinct conductance-based model neurons with discrete clusters of persistent sodium channels. Analytical solutions, expressed in terms of a Volterra series expansion for the voltage in response to a suprathreshold input current at the soma of one neuron, are obtained to investigate dendritic spikes, and the effect of backpropagation on distal dendritic spike-like potentials.

Numerical simulation of excitation-contraction coupling in a locus of the small bowel

A mathematical model for the excitation-contraction coupling within a functional unit (locus) of the small bowel is proposed. The model assumes that: the functional unit is an electromyogenic syncytium; its electrical activity is defined by kinetics of L- and T-type Ca(2+)-channels, mixed Ca(2+)-dependent K(+)-channels, potential-sensitive K(-)channels and Cl(-)-channels; the basic neural circuit, represented by the cholinergic and adrenergic neurones, provides a regulatory input to the functional unit via receptor-linked L-type Ca(2+)-channels; the smooth muscle syncytium of the locus is a null-dimensional contractile system. With the proposed model the dynamics of active force generation is determined entirely by the concentration of cytosolic calcium. The model describes electrical processes of the propagation of excitation along the neural circuit, chemical mechanisms of nerve-pulse transmission at the synaptic zones and the dynamics of active force generation. Numerical simulations have shown that it is capable of displaying different electrical patterns and mechanical responses of the locus. The simulated effects of tetrodotoxin, beta-bungarotoxin, salts of divalent cations, inhibitors of catechol-O-methyltransferase and neuronal uptake mechanisms, and changes in the concentration of external Ca2+ on the dynamics of force generation have been analysed. The results are in good qualitative and quantitative agreement with results of experiments conducted on the visceral smooth muscle of the small bowel.

Model predictions of myoelectrical activity of the small bowel

A mathematical model for the periodic electrical activity of a functional unit of the small intestine is developed. Based on real morphological and electrophysiological data, the model assumes that: the functional unit is an electromyogenic syncytium; the kinetics of L, T-type Ca2+, mixed Ca(2+)-dependent K+, potential sensitive K+ and Cl- channels determines electrical activity of the functional unit; the basic neural circuit, represented by a single cholinergic neurone, provides an excitatory input to the functional unit via receptor-linked L-type Ca2+ channels. Numerical simulation of the model has shown that it is capable of displaying the slow waves and that slight modifications of some of the parameters result in different electrical responses. The effects of the variations of the main parameters have been analyzed for their ability to reproduce various electrical patterns. The results are in good qualitative and quantitative agreement with results of experiments conducted on the small intestine.