/* * From Spikers To Bursters Via Heterogeneity * * Model Status * * This model integrates in OpenCell to produce an oscillating * output but this output has not been checked against the publication. * * ValidateCellML confirms this model as valid CellML but detects * unit inconsistencies. We have checked this model and the unit * inconsistency appears to be inherent in the publication. Attempting * to balance the units by changing the units of tau from milliseconds * to picofarads results in a broken simulation output. * * Model Structure * * When exposed to a threshold concentration of glucose, pancreatic * beta-cells exhibit a complicated pattern of electrical activity. * Bursts of action potential spikes (the "active" phase) are observed, * separated by a "silent" phase of membrane repolarisation. At * even higher glucose concentrations, continuous action potentials * are seen. This electrical activity has two important physiological * correlates: increased cytosolic Ca2+ concentration ([Ca2+]i) * and increased rate of insulin secretion during the active phase. * It is generally accepted that the rise in [Ca2+]i plays a major * role in insulin secretion and that the action potential spikes * during a burst are responsible for the rise in [Ca2+]i. * * Normal bursting patterns are only observed when the beta-cells * act synchronously, as they do in vivo, in electrically coupled * organs called the islets of Langerhans. Isolated spiking cells, * incapable of bursting by themselves, may generate islet-like * bursting rhythms when coupled. There are three alternative hypotheses * to explain why single cells and islets are different. One of * these emphasises heterogeneity: single cell properties are variable, * and individual cells may fall outside the narrow parameter regime * required for bursting. By contrast, in coupled populations, * cells with an excess of current can be balanced by cells with * too little, and the population is more likely to achieve bursting. * A second hypothesis focuses on noise from stochastic channel * fluctuations in individual cells preventing bursting by causing * premature transitions between active and silent phases. However, * in large populations, fluctuations are buffered, allowing the * underlying burst dynamics to be dominant. A third alternative * discusses the paracrine effects of glucagon secreted by alpha-cells * in the islet, on beta-cells. * * In their 2001 paper, Gerda De Vries and Arthur Sherman contrast * the first two hypotheses. They introduce a mathematical model * for bursting in pancreatic beta-cells, which is a simplified * version of the biophysically based, earlier model by Sherman * and Rinzel (1992). This model describes three ionic currents * (see below): a fast voltage-activated calcium current, ICa; * a delayed rectifier potassium current, IK; and a very slow inhibitory * potassium current, Is. The calcium and delayed rectifier potassium * currents are responsible for generating action potentials. The * slow potassium current plays no essential role in the individual * cells but heterogeneity among the cells is generated by varying * the value of the parameter beta. Parameter values were chosen * such that the individual cells are incapable of bursting. De * Vries and Sherman use their model to show bifurcation in a two-cell * system, and they extrapolate this for larger clusters of coupled * cells. * * The complete original paper reference is cited below: * * From Spikers to Bursters Via Coupling: Help From Heterogeneity, * Gerda De Vries and Arthur Sherman, 2001, Bulletin of Mathematical * Biology , 63, 371-391. PubMed ID: 11276531 * * The raw CellML description of the model can be downloaded in * various formats as described in * * diagram of the model * * [[Image file: devries_2001.png]] * * A schematic representation of the three transmembrane currents * captured by the De Vries and Sherman 2001 pancreatic beta-cell * model. */ import nsrunit; // Warning: unit conversion turned off due to unit errors in 1 equation(s) unit conversion off; // unit millisecond predefined // unit millivolt predefined unit nanoS=1E-9 kilogram^(-1)*meter^(-2)*second^3*ampere^2; unit picoA=1E-12 ampere^1; math main { //Warning: the following variables were set 'extern' or given // an initial value of '0' because the model would otherwise be // underdetermined: V, n, s realDomain time millisecond; time.min=0; extern time.max; extern time.delta; real V(time) millivolt; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) V=0; real tau millisecond; tau=20.0; real i_K(time) picoA; real i_Ca(time) picoA; real i_s(time) picoA; real g_Ca nanoS; g_Ca=3.6; real V_Ca millivolt; V_Ca=25.0; real m_infinity(time) dimensionless; real V_m millivolt; V_m=-20.0; real theta_m millivolt; theta_m=12.0; real V_K millivolt; V_K=-75.0; real g_K nanoS; g_K=10.0; real n(time) dimensionless; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) n=0; real n_infinity(time) dimensionless; real V_n millivolt; V_n=-16.0; real theta_n millivolt; theta_n=5.6; real lamda dimensionless; lamda=0.8; real g_s nanoS; g_s=4.0; real s(time) dimensionless; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) s=0; real s_infinity(time) dimensionless; real V_s millivolt; V_s=-38.0; real theta_s millivolt; theta_s=10.0; real beta dimensionless; beta=0.0; real tau_s millisecond; tau_s=35000.0; // // V:time=((-1)*(i_Ca+i_K+i_s)/tau); // i_Ca=(g_Ca*m_infinity*(V-V_Ca)); // m_infinity=(1/(1+exp((V_m-V)/theta_m))); // i_K=(g_K*n*(V-V_K)); // n:time=(lamda*(n_infinity-n)/tau); n_infinity=(1/(1+exp((V_n-V)/theta_n))); // i_s=(g_s*s*(V-V_K)); // s:time=((s_infinity-s+beta)/tau_s); s_infinity=(1/(1+exp((V_s-V)/theta_s))); }