Dynamical Systems in Neuroscience:
Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:
60 One-Dimensional Systems40bistability (I=0)excited40monostability (I=60)excitedmembrane potential, V (mV)200-20-40V(t)restingmembrane potential, V (mV)200-20-40V(t)-600 1 2 3 4 5time (ms)a-600 1 2 3 4 5time (ms)bFigure 3.6: Typical voltage trajectories of the I Na,p -model (3.5) having different valuesof I.membrane potential, VV 0V(0)V(h)V(2h)V(3h)V(t)=E L+(V 0-E L)e -g L t/CE Ltime, tFigure 3.7: Explicit analytical solution (V (t) = E L + (V 0 − E L )e −g Lt/C ) of the linearequation (3.1) and corresponding numerical approximation (dots) using Euler’s method(3.7).solid curve, is an explicit analytical solution to the linear equation (3.1) (check bydifferentiating).Finding explicit solutions is often impossible even for such simple systems as (3.5),so quantitative analysis is carried out mostly via numerical simulations. The simplestprocedure to solve (3.6) numerically, known as first-order Euler method, replaces (3.6)by the discretized system[V (t + h) − V (t)]/h = F (V (t))where t = 0, h, 2h, 3h, . . . , is the discrete time and h is a small time step. Knowing thecurrent state V (t), we can find the next state point viaV (t + h) = V (t) + hF (V (t)) . (3.7)Iterating this difference equation starting with V (0) = V 0 , we can approximate the
One-Dimensional Systems 61membrane potential, V (mV) graph of F(V)=V100500-50I LV>0 V=0 V
- Page 20 and 21: 10 Introductionspikingmoderesting m
- Page 22 and 23: 12 Introduction(a)spikinglimitcycle
- Page 24 and 25: 14 Introductionco-existence of rest
- Page 26 and 27: 16 Introduction1.2.4 Neuro-computat
- Page 28 and 29: 18 IntroductionspikeFigure 1.16: Ph
- Page 30 and 31: 20 Introductionwith coupled neurons
- Page 32 and 33: 22 IntroductionFigure 1.17: John Ri
- Page 34 and 35: 24 Introduction
- Page 36 and 37: 26 Electrophysiology of NeuronsInsi
- Page 38 and 39: 28 Electrophysiology of Neuronsouts
- Page 40 and 41: 30 Electrophysiology of NeuronsRest
- Page 42 and 43: 32 Electrophysiology of NeuronsFigu
- Page 44 and 45: 34 Electrophysiology of Neuronsm (V
- Page 46 and 47: 36 Electrophysiology of Neurons10.8
- Page 48 and 49: 38 Electrophysiology of Neurons2.3
- Page 50 and 51: 40 Electrophysiology of NeuronsFigu
- Page 52 and 53: 42 Electrophysiology of NeuronsDepo
- Page 54 and 55: 44 Electrophysiology of NeuronsV(x,
- Page 56 and 57: 46 Electrophysiology of Neurons1m (
- Page 58 and 59: 48 Electrophysiology of NeuronsPara
- Page 60 and 61: 50 Electrophysiology of NeuronsFigu
- Page 62 and 63: 52 Electrophysiology of Neuronsvolt
- Page 64 and 65: 54 Electrophysiology of Neurons
- Page 66 and 67: 56 One-Dimensional SystemsActivatio
- Page 68 and 69: 58 One-Dimensional Systemsinwardcur
- Page 72 and 73: 62 One-Dimensional Systemsat every
- Page 74 and 75: 64 One-Dimensional Systems?F(V)>0+
- Page 76 and 77: 66 One-Dimensional SystemsUnstableE
- Page 78 and 79: ¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡
- Page 80 and 81: 70 One-Dimensional Systems+ - - + +
- Page 82 and 83: 72 One-Dimensional Systemsλ(V-Veq)
- Page 84 and 85: 74 One-Dimensional Systemsbifurcati
- Page 86 and 87: 76 One-Dimensional Systems100806040
- Page 88 and 89: 78 One-Dimensional Systemsbifurcati
- Page 90 and 91: 80 One-Dimensional Systems20 mV100
- Page 92 and 93: 82 One-Dimensional Systemsmembrane
- Page 94 and 95: 84 One-Dimensional Systemssteady-st
- Page 96 and 97: 86 One-Dimensional Systemsspike rig
- Page 98 and 99: 88 One-Dimensional SystemsVF (V)1VV
- Page 100 and 101: 90 One-Dimensional Systems10.80.60.
- Page 102 and 103: 92 One-Dimensional Systems
- Page 104 and 105: 94 Two-Dimensional SystemsI Na,pneu
- Page 106 and 107: 96 Two-Dimensional Systemsdefines a
- Page 108 and 109: 98 Two-Dimensional Systems(f(x(t),y
- Page 110 and 111: 100 Two-Dimensional Systems0.70acti
- Page 112 and 113: 102 Two-Dimensional Systems0.70.6K
- Page 114 and 115: 104 Two-Dimensional Systemsy1.510.5
- Page 116 and 117: 106 Two-Dimensional SystemsFor exam
- Page 118 and 119: 108 Two-Dimensional SystemsIn gener
One-Dimensional <strong>Systems</strong> 61membrane potential, V (mV) graph of F(V)=V100500-50I LV>0 V=0 V
Change language