A mathematical model of the crustacean stretch receptor neuron. Biomechanics of the receptor muscle, mechanosensitive ion channels, and macrotransducer properties
Authors: Swerup C. , Rydqvist B. 
Published: December 2017

1. A mathematical model of the primary transduction process in a mechanoreceptor, the slowly adapting stretch receptor organ of the crayfish, has been developed taking into account the viscoelastic properties of the accessory structures of the receptor, i.e., the receptor muscle, the biophysical properties of the mechanosensitive channels (MSCs) and the passive electrical properties of the neuronal membrane (leak conductance and capacitative properties). The work is part of an effort to identify and characterize the mechanical and ionic mechanisms in a complex mechanoreceptor. The parameters of the model are based mainly on results of our own experiments and to some extent on results from other studies. The performance of the model has been compared with the performance of the slowly adapting receptor. 2. The model resulted in nonlinear differential equations that were solved by an iterative, fourth order Range-Kutta method. For the calculations of potential, the cell was treated as an idealized spherical body. The extension of the receptor muscle was 0-30%, which is within the physiological limits for this receptor. 3. The mechanical properties of the receptor muscle were modeled by a simple Voigt element (a spring in parallel with a dashpot) in series with a nonlinear spring. This element can describe resonably well the tension development in the receptor muscle at least for large extensions (> 12%). However, for small extensions (< 12%), the muscle seems to be more stiff than for large extensions. 4. The receptor current at different extensions of the receptor was computed using typical viscoelastic parameters for a receptor muscle together with a transformation of muscle tension to tension in the neuronal dendrites and finally the properties of the mechanosensitive channels. The model fit was satisfactory in the high extension range whereas in the low extension range the deviation from the experimental results could be explained partly by insufficient modeling of the nonlinear viscoelastic properties. The voltage dependence of the receptor current was also well predicted by the model. 5. If the parameters of the viscoelastic model were adjusted for each extension so that each tension response closely resembled the experimental values, the fit of the current responses was improved but still deviated from the experimental currents. One factor that might explain the difference is the possibility that the MSCs in the stretch receptor neuron might have intrinsic adaptive properties. Introducing an exponential adaptive behavior of individual MSCs increased the ability of the model to predict the receptor current. 6. The receptor potential was calculated by modeling the neuronal membrane by a lumped leak conductance and capacitance The calculated receptor potential was higher than the experimental receptor potential. However, the fit of the receptor potential was improved substantially by introducing an adaptation of the MSCs as outlined in the preceding paragraph. the remaining discrepancy might be explained by insufficient blocking of K+ channels in the experiment. 7. The model can predict a wide range of experimental data from the slowly adapting stretch receptor neuron including the mechanical response of the receptor muscle, the receptor current and its voltage dependence, and the receptor potential. It also describes accurately the passive electrical properties of the neuronal membrane.