/* * Modeling N-methyl-D-aspartate-induced bursting in dopamine neurons * * Model Status * * This is the more elaborate model based on the equations in the * the appendix of the paper. The CellML model runs in both COR * and OpenCell and the units are consistent, however it is unable * to recreate the figures in the published paper. * * Model Structure * * ABSTRACT: Burst firing of dopaminergic neurons of the substantia * nigra pars compacta can be induced in vitro by the glutamate * agonist N-methyl-D-aspartate. It has been suggested that the * interburst hyperpolarization is due to Na+ extrusion by a ouabain-sensitive * pump (Johnson et al. (1992) Science 258, 665-667). We formulate * and explore a theoretical model, with a minimal number of currents, * for this novel mechanism of burst generation. This minimal model * is further developed into a more elaborate model based on observations * of additional currents and hypotheses about their spatial distribution * in dopaminergic neurons (Hounsgaard (1992) Neuroscience 50, * 513-518; Llinas et al. (1984) Brain Res. 294, 127-132). Using * the minimal model, we confirm that interaction between the regenerative, * inward N-methyl-D-aspartate-mediated current and the outward * Na(+)-pump current is sufficient to generate the slow oscillation * (approximately 0.5 Hz) underlying the burst. The negative-slope * region of the N-methyl-D-aspartate channel's current-voltage * relation is indispensable for this slow rhythm generation. The * time-scale of Na(+)-handling determines the burst's slow frequency. * Moreover, we show that, given the constraints of sodium handling, * such bursting is best explained mechanistically by using at * least two spatial, cable-like compartments: a soma where action * potentials are produced and a dendritic compartment where the * slow rhythm is generated. Our result is consistent with recent * experimental evidence that burst generation originates in distal * dendrites (Seutin et al. (1994) Neuroscience 58, 201-206). Responses * of the model to a number of electrophysiological and pharmacological * stimuli are consistent with known responses observed under similar * conditions. These include the persistence of the slow rhythm * when the tetrodotoxin-sensitive Na+ channel is blocked and when * the soma is voltage-clamped at -60 mV. Using our more elaborate * model, we account for details of the observed frequency adaptation * in N-methyl-D-aspartate-induced bursting, the origin of multiple * spiking and bursting mechanisms, and the interaction between * two different bursting mechanisms. Besides reproducing several * well established firing patterns, this model also suggests that * new firing modes, not yet recorded, might also occur in dopaminergic * neurons. This model provides mechanistic insights and explanations * into the origin of a variety of experimentally observed membrane * potential firing patterns in dopaminergic neurons, including * N-methyl-D-aspartate-induced bursting and its dendritic origin. * Such a model, capable of reproducing a number of realistic behaviors * of dopaminergic neurons, could be useful in further studies * of the basal ganglia-thalamocortical motor circuit. It may also * shed light on bursting that involves N-methyl-D-aspartate channel * activity in other neuron types. * * The original paper reference is cited below. * * Modeling N-methyl-D-aspartate-induced bursting in dopamine neurons, * Y.X. Li, R. Bertram, and J. Rinzel, 1996, Neuroscience, 71, * 397-410. PubMed ID: 9053795 * * cell schematic for the model * * [[Image file: li_1996.png]] * * Schematic representation of the minimal model of a DA neuron. * It consists of a spike-producing compartment(soma) and a slow * rhythm-generating compartment (lumped dendrite). The two compartments * are electrotonically coupled with coupling conductance gc. The * types of ionic channels considered are just sufficient for a * qualitative understanding of the mechanism underlying NMDA-induced * bursting in DA neurons. No voltage-dependent channels other * than the NMDA channel are considered in the dendrite in this * minimal model. */ import nsrunit; unit conversion on; unit millisecond_per_second=.001 dimensionless; unit per_second=1 second^(-1); unit mM=1 meter^(-3)*mole^1; unit uM=1E-3 meter^(-3)*mole^1; unit mV=.001 kilogram^1*meter^2*second^(-3)*ampere^(-1); unit cm2=1E-4 meter^2; unit uF=1E-6 kilogram^(-1)*meter^(-2)*second^4*ampere^2; unit uA=1E-6 ampere^1; unit nA=1E-9 ampere^1; unit uF_per_cm2=.01 kilogram^(-1)*meter^(-4)*second^4*ampere^2; unit mS_per_cm2=10 kilogram^(-1)*meter^(-4)*second^3*ampere^2; unit uA_per_cm2=.01 meter^(-2)*ampere^1; unit mMcm2_per_uAs=100 meter^(-1)*second^(-1)*ampere^(-1)*mole^1; math main { realDomain time second; time.min=0; extern time.max; extern time.delta; real V_s(time) mV; when(time=time.min) V_s=-64.0; real V_Na mV; V_Na=55; real I_Na_s(time) uA_per_cm2; real I_K_DR_s(time) uA_per_cm2; real I_Ca_T(time) uA_per_cm2; real I_K_Ca(time) uA_per_cm2; real I_A(time) uA_per_cm2; real I_h(time) uA_per_cm2; real g_Na_s mS_per_cm2; g_Na_s=3.2; real g_K_DR_s mS_per_cm2; g_K_DR_s=6.4; real g_Ca_T mS_per_cm2; g_Ca_T=1.5; real g_K_Ca mS_per_cm2; g_K_Ca=1.2; real g_A mS_per_cm2; g_A=2; real g_h mS_per_cm2; g_h=0.1; real Ca(time) mM; when(time=time.min) Ca=0; real k_Ca per_second; k_Ca=1; real K_Ca mM; K_Ca=0.0004; real V_h mV; V_h=-30; real beta mMcm2_per_uAs; beta=0.104; real I_APP uA_per_cm2; I_APP=-6.7; real V_Ca mV; V_Ca=120; real V_K mV; V_K=-85; real n(time) dimensionless; when(time=time.min) n=0.002; real h(time) dimensionless; when(time=time.min) h=1.0; real m_infinity(time) dimensionless; real g_c mS_per_cm2; g_c=0.1; real p dimensionless; p=0.5; real V_D(time) mV; when(time=time.min) V_D=-77.0; real C_m uF_per_cm2; C_m=1; real m_T(time) dimensionless; when(time=time.min) m_T=0.1; real h_T(time) dimensionless; when(time=time.min) h_T=0.1; real a(time) dimensionless; when(time=time.min) a=0.1; real b(time) dimensionless; when(time=time.min) b=0.1; real m_h(time) dimensionless; when(time=time.min) m_h=0.1; real Na(time) mM; when(time=time.min) Na=5.09; real K_p mM; K_p=15; real I_L(time) uA_per_cm2; real I_pump(time) uA_per_cm2; real R_pump uA_per_cm2; R_pump=18; real Na_eq mM; Na_eq=8; real phi_Na(time) dimensionless; real phi_Na_eq dimensionless; real alpha mMcm2_per_uAs; alpha=0.173; real I_NMDA(time) uA_per_cm2; real I_Na_NMDA(time) uA_per_cm2; real g_NMDA mS_per_cm2; g_NMDA=25; real g_Na_NMDA mS_per_cm2; g_Na_NMDA=5; real g_L mS_per_cm2; g_L=0.18; real Mg_o mM; Mg_o=1.4; real K_Mg mM; K_Mg=10; real q mV; q=12.5; real V_NMDA mV; V_NMDA=0; real V_L mV; V_L=-50; real I_D(time) uA_per_cm2; real I_Ca_L(time) uA_per_cm2; real g_Ca_L mS_per_cm2; g_Ca_L=0.19; real g_K_DR_D mS_per_cm2; g_K_DR_D=0.14; real I_K_DR_D(time) uA_per_cm2; real m_L(time) dimensionless; when(time=time.min) m_L=0.1; real m_T_infinity(time) dimensionless; real h_T_infinity(time) dimensionless; real a_infinity(time) dimensionless; real b_infinity(time) dimensionless; real m_h_infinity(time) dimensionless; real m_L_infinity(time) dimensionless; real n_infinity(time) dimensionless; real h_infinity(time) dimensionless; real tau_h(time) second; real tau_n(time) second; real tau_m_T second; real tau_h_T second; real tau_m_L(time) second; real tau_a second; real tau_b second; real tau_m_h second; // // I_Na_s=(g_Na_s*h*(V_s-V_Na)*m_infinity^3); I_K_DR_s=(g_K_DR_s*(V_s-V_K)*n^2); I_Ca_T=(g_Ca_T*h_T*(V_s-V_Ca)*m_T^2); I_K_Ca=(g_K_Ca*Ca^4/(Ca^4+K_Ca^4)*(V_s-V_K)); I_A=(g_A*b*(V_s-V_K)*a^4); I_h=(g_h*m_h*(V_s-V_h)); V_s:time=((1E3 millisecond_per_second)*(I_APP-(I_Na_s+I_Ca_T+I_K_DR_s+I_K_Ca+I_A+I_h+g_c/p*(V_s-V_D)))/C_m); Ca:time=((-1)*(beta*I_Ca_T+k_Ca*Ca)); // I_K_DR_D=(g_K_DR_D*(V_D-V_K)*n^2); phi_Na=(Na^3/(Na^3+K_p^3)); phi_Na_eq=(Na_eq^3/(Na_eq^3+K_p^3)); I_pump=(R_pump*(phi_Na-phi_Na_eq)); I_L=(g_L*(V_D-V_L)); I_NMDA=(g_NMDA/(1+Mg_o/K_Mg*exp((-1)*V_D/q))*(V_D-V_NMDA)); I_Na_NMDA=(g_Na_NMDA/(1+Mg_o/K_Mg*exp((-1)*V_D/q))*(V_D-V_Na)); I_D=(I_NMDA+I_pump+I_L); I_Ca_L=(g_Ca_L*(V_D-V_Ca)*m_L^2); V_D:time=((-1)*((1E3 millisecond_per_second)*(I_Ca_L+I_K_DR_D+I_NMDA+I_pump+I_L+g_c/(1-p)*(V_D-V_s)))/C_m); Na:time=(alpha*((-1)*I_Na_NMDA-I_pump*3)); // // m_infinity=(1/(1+exp((-1)*(V_s+(35 mV))/(6.2 mV)))); n_infinity=(1/(1+exp((-1)*(V_s+(31 mV))/(5.3 mV)))); h_infinity=(1/(1+exp((V_s+(30 mV))/(8.3 mV)))); m_T_infinity=(1/(1+exp((-1)*(V_s+(55 mV))/(7 mV)))); h_T_infinity=(1/(1+exp((V_s+(81 mV))/(11 mV)))); a_infinity=(1/(1+exp((-1)*(V_s+(60 mV))/(10 mV)))); b_infinity=(1/(1+exp((V_s+(70 mV))/(5.7 mV)))); m_h_infinity=(1/(1+exp((V_s+(80 mV))/(8 mV)))); m_L_infinity=(1/(1+exp((-1)*(V_D+(20 mV))/(5.3 mV)))); tau_h=((.4 second)*(1+2/(1+exp((V_s+(25 mV))/(5 mV))))); tau_n=((.8 second)*(1+2/(1+exp((V_s+(25 mV))/(10 mV))))/(1+exp((-1)*(V_s+(70 mV))/(10 mV)))); tau_m_L=((.4 second)/(5*exp((-1)*(V_D+(11 mV))/(8.3 mV))+(-1)*(V_D+(11 mV))/(8.3 mV)/(exp((-1)*(V_D+(11 mV))/(8.3 mV))-1))); tau_m_T=(1 second); tau_h_T=(10 second); tau_a=(.5 second); tau_b=(10 second); tau_m_h=(190 second); h:time=((h_infinity-h)/tau_h); n:time=((n_infinity-n)/tau_n); m_T:time=((m_T_infinity-m_T)/tau_m_T); m_L:time=((m_L_infinity-m_L)/tau_m_L); m_h:time=((m_h_infinity-m_h)/tau_m_h); a:time=((a_infinity-a)/tau_a); b:time=((b_infinity-b)/tau_b); h_T:time=((h_T_infinity-h_T)/tau_h_T); }