混合尖峰发放模型,To understand how the brain works, we need to combine experimentalstudies of animal and human nervous systems with numerical simulation of large-scale brain models. E.M. Izhikevich,神经元是生物体中信息传递的基本单元。,文章解读（Hybrid spiking models） several kinds of modelssimulation methods of simple model of spiking neurons 问题探究重现图2中不同神经元的活动模式，并指出 相应活动模式的参数统计各种模式下发放率与输入刺激的强度之 间的关系输入均值为 符合泊松分布的随机电脉冲， 考察均值与发放率之间关系,The leaky integrate-and-fire model,v:membrane potentialC:memberane capacitancegleak:leaky ohmic conductanceEleak:leaky reverse potentialI:input current,It lacks an intrinsic spike-generation mechanism, and hence is just a threshold model.,The quadratic integrate-and-fire model,v:membrane potentialC:memberane capacitanceI:input currentK:a positive parameterVrest:resting potentialsVthresh: instantaneous threshold potentials,A hybrid spiking modelas it combines a smooth spike-generation mechanism with autocatalytic upstroke of the spike and a hard after-spike reset.,Simple model of spiking neurons,U:a recovery variable a: the time scale of u b: the sensitivity of u c: the after-spike reset value of v d: the after-spike reset of u,Simple model of spiking neurons,U:a recovery variable a: the time scale of u b: the sensitivity of u c: the after-spike reset value of v d: the after-spike reset of u,Other hybrid spiking models,All after-spiking resetting are the same.,U：a conductanceE:the reverse potential of u,Simple model of spiking neurons Simulation methods,1)How to detect Vpeak crossing? 2)Strong synaptic conductances cause numerical instability. How to avoid numerical instability?,Simple model of spiking neurons Simulation methods,Detecting Vpeak crossing:,linear interpolation,Avoiding numerical instability:,Simple model of spiking neurons Simulation methods,time-varying conductance,the reverse potential for a particular synaptic current i e.g. i = NMDA, AMPA, GABAA and GABAB,Avoiding numerical instability:,Simple model of spiking neurons Simulation methods,The total conductanceThe total reverse potential,Avoiding numerical instability:,Simple model of spiking neurons Simulation methods,Indeed, positive values of g(t) push the membrane voltage towards the reverse potential E(t). However, when g(t) is large, the term tg(t)E(t) v(t) becomes large, resulting in overshoot and divergence from E(t). Notice that, no matter how small the simulation step t is, the conductance variable could always become so large as to create the instability.,Avoiding numerical instability:,Simple model of spiking neurons Simulation methods,Finally，,Since the numerical instability is caused by the linear term, one can use the hybrid numerical method that combines the simplicity and efficiency of explicit methods and the numerical stability of implicit methods.,Avoiding numerical instability:,Simple model of spiking neurons Simulation methods,Left: Right:,问题探究一,重现图2中不同神经元的活动模式，并指出相应活动模式的参数 form “Simple model of spiking neurons”,问题探究一:程序示例,subplot(5,4,1) a=0.02; b=0.2; c=-65; d=6; V=-70; u=b*V; VV=; uu=; tau = 0.25; tspan = 0:tau:100; T1=tspan(end)/10; for t=tspanif (tT1) I=14;elseI=0;end;V = V + tau*(0.04*V2+5*V+140-u+I);u = u + tau*a*(b*V-u);,if V 30VV(end+1)=30;V = c;u = u + d;elseVV(end+1)=V;end;uu(end+1)=u; end; plot(tspan,VV,0 T1 T1 max(tspan),-90+0 0 10 10); axis(0 max(tspan) -90 30) axis off; title(A) tonic spiking);,问题探究一:figure 1,问题探究一:figure 2,问题探究一:参数,问题探究一:参数,问题探究二,统计各种模式下发放率和输入刺激的强度之间的关系,1）在不同模式下，改变输入刺激电流I的大小，并统计不同的输入电流I对应的发放率r，利用plot函数绘制不同模式下的“r-I”关系图 2）对A模式（tonic spiking）的“r-I”关系图进行拟合，得出其对应表达式注：统计发放率方法众多，此处采用较为简便的“时间平均法”,问题探究二：程序示例,V = V + tau*(0.04*V2+5*V+140-u+I);u = u + tau*a*(b*V-u);if V 30VV(end+1)=30;V = c;u = u + d;elseVV(end+1)=V;end;uu(end+1)=u; end; ffl(i)=histc(VV,30)/100; end figure plot(dl,ffl,-o) title(A) tonic spiking);,subplot(5,4,1) a=0.02; b=0.2; c=-65; d=6; V=-70; u=b*V; VV=; uu=; tau = 0.25; tspan = 0:tau:100; T1=tspan(end)/10; dl=1:50; ffl=zeros(1,length(dl); for i=1:length(dl) for t=tspanif (tT1) I=dl(i);elseI=0;end;,问题探究二：“r-I”关系图,问题探究二,对A模式（tonic spiking）的“r-I”关系图进行拟合,Linear model Poly2:f(x) = p1*x2 + p2*x + p3 Coefficients (with 95% confidence bounds):p1 = 0.001364 (0.001358, 0.001369)p2 = 6.425e-005 (-0.0002163, 0.0003448)p3 = -0.007935 (-0.01104, -0.004833),即：,问题探究三,输入均值为 符合泊松分布的随机电脉冲，考察均值和发放率之间的关系,(1)选择“tonic spiking”的脉冲为基本脉冲单元，在一个单位的时间内有n个基本脉冲单元依次输入（n服从均值为u的泊松分布），则在此单位时间内的总输入为n个基本脉冲单元的累加总电流。 如右图例示（n=5）,问题探究二,(2)当u=5时，得出 发放率与输入刺激I的关系如右图所示：,问题探究二,(3)对于不同的u值，分别统计发放率，最终得到“r-u”关系图：,THANK YOU !,