Chapter 3. Neuronal Dynamics 2 Activation Models 3. 1 Neuronal dynamical system Neuronal activations change with time. The way they change depends on the dynamical equations as followIng x=g(Fx, Fy, (3-1) y=h(FX,Fy,…) (3-2) 2002.10.8
2002.10.8 Chapter 3. Neuronal Dynamics 2 :Activation Models Neuronal activations change with time. The way they change depends on the dynamical equations as following: x = g（FX ，FY ，） • y = h（FX ，FY ，） • (3-1) (3-2) 3.1 Neuronal dynamical system
3. 1 ADDITIVE NEURONAL DYNAMICS a first-order passive decay model In the absence of external or neuronal stimuli the simplest activation dynamics model is Z (3-3) (3-4) 2002.10.8
2002.10.8 In the absence of external or neuronal stimuli, the simplest activation dynamics model is: i xi = −x • y j = −yj • 3.1 ADDITIVE NEURONAL DYNAMICS (3-3) (3-4) first-order passive decay model
3.1 ADDITIVE NEURONAL DYNAMICS n since for any finite initial condition (t)=x1(0)e The membrane potential decays exponentially quickly to its zero potential
t xi t xi e − ( ) = (0) since for any finite initial condition 3.1 ADDITIVE NEURONAL DYNAMICS The membrane potential decays exponentially quickly to its zero potential
Passive membrane decay assive-decay rate A.>0 scales the rate to the membrane 's resting potential solution: X(=X(0)6 Passive-decay rate measures the cell membrane's resistance or“ friction” to current flow 2002.10.8
2002.10.8 Passive Membrane Decay Passive-decay rate Ai 0 scales the rate to the membrane’s resting potential. solution : At i i i x t x e − （ ）= （0） Passive-decay rate measures: the cell membrane’s resistance or “friction” to current flow. i i i x = −A x •
property Pay attention to A property The larger the passive-decay rate the faster the decay--the less the resistance to current flow
Ai The larger the passive-decay rate,the faster the decay--the less the resistance to current flow. Pay attention to property property
Membrane time Constants The membrane time constant c scales the time variable of the activation dynamical system The multiplicative constant model A (3-8) 2002.10.8
2002.10.8 Membrane Time Constants The membrane time constant scales the time variable of the activation dynamical system. The multiplicative constant model: Ci i i i xi C x = -A • （3－8）
Solution and property solution X()=x1(0)e I property The smaller the capacitance, the faster things change As the membrane capacitance increases toward positive infinity, membrane fluctuation slows to stop
Solution and property solution t C A i i i t x e − x ( ) = (0) i property The smaller the capacitance ,the faster things change As the membrane capacitance increases toward positive infinity,membrane fluctuation slows to stop
Membrane resting potentials Definition Define resting potential p as the activation value to which the membrane potential equilibrates in the absence of external or neuronal inputs a x: +p (3-11) Solutions x;(t)=x;(0e+1(1-e) (3-12) A 2002.10.8
2002.10.8 Membrane Resting Potentials Define resting Potential as the activation value to which the membrane potential equilibrates in the absence of external or neuronal inputs: Pi i i i i Ci x = -A x + P • Solutions (1- e ) A P x (t) x (0)e t C A - i i t C A - i i i i i i = + Definition (3-11) (3-12)
Note The capacitance appear in the index of the solution it is called time-scaling capacitance It does no affect the steady-state solution and does not depend on the finite initial condition In resting case we can find the solution quickly
Note The capacitance appear in the index of the solution,it is called time-scaling capacitance. It does no affect the steady-state solution and does not depend on the finite initial condition. In resting case,we can find the solution quickly
Additive External Input Add input Apply a relatively constant numeral input to a neuron Xi=-X;+1 (3-13) solution X(t)=x:(0)e+l1(1-e)(3-14)
Additive External Input Add input Apply a relatively constant numeral input to a neuron. i i xi = -x + I • solution x (t) x (0)e I (1- e ) -t i -t i = i + (3-13) (3-14)