Dynamical Systems in Neuroscience:
Dynamical Systems in Neuroscience: Dynamical Systems in Neuroscience:
76 One-Dimensional Systems100806040200100806040200100806040200bistabilityF(V)rest thresholdbifurcationF(V)tangent pointmonostabilityF(V)-50 0 50membrane potential, V (mV)I=0I=16I=60membrane potential, V (mV) membrane potential, V (mV) membrane potential, V (mV)40200-20-40-6040200-20-40-6040200-20-40V(t)excited statethresholdrestV(t)excited stateV(t)excited state-600 1 2 3 4 5time (ms)Figure 3.25: Bifurcation in the I Na,p -model (3.5): The rest state and the thresholdstate coalesce and disappear when the parameter I increases.
One-Dimensional Systems 77F(V)I=18I=17I=16I=15I=14I=13I=12I=11no equilibriabifurcationtwo equilibriatangentpointVstable equilibriaunstable equilibriaFigure 3.26: Saddle-node bifurcation: As the graph of the function F (V ) is lifted up,the stable and unstable equilibria approach each other, coalesce at the tangent point,and then disappear.saddle-nodenot saddle-nodeFVnon-hyperbolichyperbolichyperbolicnon-degenerate degenerate degeneratetransversalnot transversalnot transversalFigure 3.27: Geometrical illustration of the three conditions defining saddle-node bifurcations.Arrows denote the direction of displacement of the function F (V, I) as thebifurcation parameter I changes.
- 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 70 and 71: 60 One-Dimensional Systems40bistabi
- 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 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
- Page 120 and 121: 110 Two-Dimensional Systemsv 2 v 2v
- Page 122 and 123: 112 Two-Dimensional SystemsV-nullcl
- Page 124 and 125: 114 Two-Dimensional SystemsV-nullcl
- Page 126 and 127: 116 Two-Dimensional Systemsheterocl
- Page 128 and 129: 118 Two-Dimensional Systems0.60.5n-
- Page 130 and 131: 120 Two-Dimensional Systemseigenval
- Page 132 and 133: 122 Two-Dimensional SystemsK + acti
- Page 134 and 135: 124 Two-Dimensional Systemsexcitati
76 One-Dimensional <strong>Systems</strong>100806040200100806040200100806040200bistabilityF(V)rest thresholdbifurcationF(V)tangent po<strong>in</strong>tmonostabilityF(V)-50 0 50membrane potential, V (mV)I=0I=16I=60membrane potential, V (mV) membrane potential, V (mV) membrane potential, V (mV)40200-20-40-6040200-20-40-6040200-20-40V(t)excited statethresholdrestV(t)excited stateV(t)excited state-600 1 2 3 4 5time (ms)Figure 3.25: Bifurcation <strong>in</strong> the I Na,p -model (3.5): The rest state and the thresholdstate coalesce and disappear when the parameter I <strong>in</strong>creases.
Change language