/* * Numerical simulation of motility patterns of the small bowel. * 1. formulation of a mathematical model * * Model Status * * This CellML model represents the smooth muscle cell described * in the published paper (equations 20-27). The model runs in * both OpenCell and COR and the units are consistent. However * the model does not replicate the results in figure 2 of the * paper - this is most likely to be due to a lack of a stimulus * input. The original paper describes several cell types which * are linked together, and a mechanical stimulus which is detected * by a neuron (eq. 31). Therefore to get this model to run correctly * the other cell models will have to be coded up and linked together. * CellML 1.1 is suited for this purpose. * * Model Structure * * ABSTRACT: A complete mathematical model of the periodic myoelectrical * activity of a functional unit of the small intestine is presented. * Based on real morphological and electrophysiological data, the * model assumes that: the functional unit is an electromyogenic * syncytium; the kinetics of L-type Ca2+, T-type Ca2+, Ca2+-activated * K+, voltage dependent K+and Cl-channels determine the electrical * activity of the functional unit; the enteric nervous system * is satisfactorily represented by an efferent cholinergic neuron * that provides an excitatory input to the functional unit through * receptor-linked L-type Ca2+channels and by an afferent pathway * composed of the primary and secondary sensory neurons; the dynamics * of propagation of the wave of depolarization along the unmyelinated * nerve axons satisfy the Hodgkin-Huxley model; the electrical * activity of the neural soma reflects the interaction of N-type * Ca2+channels, Ca2+-activated K+and voltage dependent Na+, K+and * Cl-channels; the smooth muscle syncytium of the locus is a null-dimensional * contractile system. With the proposed model the dynamics of * active force generation are determined entirely by the concentration * of cytosolic calcium. The model describes: the mechanical excitation * of the free nerve endings of the mechanoreceptor of the receptive * field of the pathway; the electrical processes of the propagation * of excitation along the afferent and efferent neural circuits; * the chemical mechanisms of nerve-pulse transmission at the synaptic * zones; the slow wave and bursting type electrical activity; * cytosolic calcium concentration; the dynamics of active force * generation. Numerical simulations have shown that the model * can display different electrical patterns and mechanical responses * of the locus. The results show good qualitative and quantitative * agreement with the results of experiments conducted on the small * intestine. * * The original paper reference is cited below: * * Numerical Simulation of Motility Patterns of the Small Bowel. * 1. Formulation of a Mathematical Model, R. N. Miftakhov, G. * R. Abdusheva and J. Christensen, 1999, Journal of Theoretical * Biology, 197, 89-112. PubMed ID: 10036210 * * cell_diagram * * [[Image file: miftakhov_1999.png]] * * The myoelectrical activity of the gastrointestinal smooth muscle * cell is governed by the dynamics of voltage-dependent Ca2+ L * and T-type channels (ICa,L and ICa,T), a voltage-gated K+ channel * (IK), a calcium-activated K+ channel (ICa-K), and a leak chloride * current (ICl). */ import nsrunit; unit conversion on; // unit millisecond predefined unit per_millisecond=1E3 second^(-1); // unit millivolt predefined unit per_millivolt=1E3 kilogram^(-1)*meter^(-2)*second^3*ampere^1; unit per_millivolt_millisecond=1E6 kilogram^(-1)*meter^(-2)*second^2*ampere^1; unit milliS_per_cm2=10 kilogram^(-1)*meter^(-4)*second^3*ampere^2; unit microF_per_cm2=.01 kilogram^(-1)*meter^(-4)*second^4*ampere^2; // unit millimolar predefined unit millimolar_per_millivolt=1E3 kilogram^(-1)*meter^(-5)*second^3*ampere^1*mole^1; unit microA_per_cm2=.01 meter^(-2)*ampere^1; math main { realDomain time millisecond; time.min=0; extern time.max; extern time.delta; real alpha dimensionless; alpha=0.12; real lamda dimensionless; lamda=12.5; real V(time) millivolt; when(time=time.min) V=-55.0; real Cm microF_per_cm2; Cm=2.5; real i_Ca_T(time) microA_per_cm2; real i_Ca_L(time) microA_per_cm2; real i_Ca_K(time) microA_per_cm2; real i_K(time) microA_per_cm2; real i_Cl(time) microA_per_cm2; real V_tilde(time) millivolt; real E_Ca millivolt; E_Ca=80.0; real g_Ca_T milliS_per_cm2; g_Ca_T=0.51; real m(time) dimensionless; real h(time) dimensionless; when(time=time.min) h=0.01; real alpha_m(time) per_millisecond; real beta_m(time) per_millisecond; real alpha_h(time) per_millisecond; real beta_h(time) per_millisecond; real g_Ca_L milliS_per_cm2; g_Ca_L=0.004; real x_Ca(time) dimensionless; when(time=time.min) x_Ca=0.01; real tau_x_Ca millisecond; tau_x_Ca=500.0; real E_K millivolt; E_K=-75.0; real g_K milliS_per_cm2; g_K=0.3; real n(time) dimensionless; when(time=time.min) n=0.01; real alpha_n(time) per_millisecond; real beta_n(time) per_millisecond; real Ca(time) millimolar; when(time=time.min) Ca=1E-4; real g_Ca_K milliS_per_cm2; g_Ca_K=0.03; real rho per_millisecond; rho=0.125E3; real K_c millimolar_per_millivolt; K_c=425.0E-5; real g_Cl milliS_per_cm2; g_Cl=0.003; real E_Cl millivolt; E_Cl=-40.0; // // // V:time=((-1)*(1/(Cm*alpha))*(i_Ca_T+i_Ca_L+i_Ca_K+i_K+i_Cl)); // V_tilde=((127*V+(8265 millivolt))/105); // i_Ca_T=(g_Ca_T*m^3*h*(V-E_Ca)); // alpha_m=((.1 per_millivolt_millisecond)*((50 millivolt)-V_tilde)/(exp(5-V_tilde*(.1 per_millivolt))-1)); beta_m=((4 per_millisecond)*exp(((25 millivolt)-V_tilde)/(18 millivolt))); m=(alpha_m/(alpha_m+beta_m)); // alpha_h=((.07 per_millisecond)*exp(((25 millivolt)-V_tilde)/(20 millivolt))); beta_h=((1 per_millisecond)/(1+exp(5.5-V_tilde*(.1 per_millivolt)))); h:time=((alpha_h*(1-h)-beta_h*h)/(alpha*lamda)); // i_Ca_L=(g_Ca_L*x_Ca*(V-E_Ca)); // x_Ca:time=((1/(1+exp((.15 per_millivolt)*((-1)*V_tilde-(50 millivolt))))-x_Ca)/(alpha*tau_x_Ca)); // i_K=(g_K*n^4*(V-E_K)); // alpha_n=((.01 per_millivolt_millisecond)*((55 millivolt)-V_tilde)/(exp(((55 millivolt)-V_tilde)/(10 millivolt))-1)); beta_n=((.125 per_millisecond)*exp(((45 millivolt)-V_tilde)/(80 millivolt))); n:time=((alpha_n*(1-n)-beta_n*n)/(alpha*lamda)); // i_Ca_K=(g_Ca_K*Ca*(V-E_K)/((.5 millimolar)+Ca)); Ca:time=(rho/alpha*(K_c*x_Ca*(E_Ca-V)-Ca)); // i_Cl=(g_Cl*(V-E_Cl)); }