Dynamical Systems in Neuroscience:

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242 ExcitabilityThe existence of fast subthreshold oscillatory potentials is a distinguishable featureof neurons near Andronov-Hopf bifurcation. Indeed, the resting state of such a neuronis a stable focus. When stimulated by a brief synaptic input or an injected pulse ofcurrent, the state of the system deviates from the focus equilibrium, and then returnsto the equilibrium along a spiral trajectory, as depicted in Fig. 7.16, top, therebyproducing a damped oscillation. The frequency of such an oscillation is the imaginarypart of the complex-conjugate eigenvalues at the equilibrium (see Sect. 6.1.3), and itcan be as large as 200 Hz in mammalian neurons.In Ex. 3 we prove that noise can make such oscillations sustained. While the stateof the system is perturbed and returns to the focus equilibrium, another strong randomperturbation may push it away from the equilibrium, thereby starting a new dampedoscillation. As a result, persistent noisy perturbations create a random sequence ofdamped oscillations and do not let the neuron rest. The membrane potential of sucha neuron exhibits noisy sustained oscillations of small amplitude depicted in Fig. 7.16and discussed in Sect. 6.1.4.Injected dc-current or background synaptic noise increase the rest potential, changeits eigenvalues and, hence change the frequency and amplitude of noisy oscillations.Fig. 7.16 depicts typical cases when the frequency and the amplitude increase as theresting state becomes more depolarized.One should be careful to distinguish fast and slow subthreshold oscillations of membranepotential. Fast oscillations, as in Fig. 7.16, are those having period comparablewith the membrane time constant or with the period of repetitive spiking. In contrast,some neurons found in entorhinal cortex, inferior olive, hippocampus, thalamus, andmany other brain regions can exhibit slow subthreshold oscillations with the periodof 100 ms and more. These oscillations reflect the interplay between fast and slowmembrane currents, e.g., I h or I T , and may be irrelevant to the bifurcation mechanismof excitability. We will discuss this issue in detail in Sect. 7.3.3 and in Chap. 9. Amazingly,such neurons still possess many neuro-computational properties of resonators,such as frequency preference and rebound spiking, but exhibit these properties on aslower time scale.7.2.2 Frequency preference and resonanceA standard experimental procedure to test the propensity of a neuron to subthresholdoscillations is to stimulate it with a sinusoidal current having slowly increasing frequency,called a zap current, as in Fig. 7.17. The amplitude of the evoked oscillationsof the membrane potential, normalized by the amplitude of stimulating oscillatory current,is called the neuronal impedance — a frequency-domain extension of the conceptof resistance. The impedance profile of integrators is decreasing while that of resonatorshas a peak corresponding to the frequency of subthreshold oscillations, around140 Hz in the mesV neuron in the figure. Thus, integrators act as low-pass filters whileresonators act as band-pass filters to periodic signals.Instead of sinusoidal stimulation, consider more biological stimulation with pulses
Excitability 243membrane potential response, mV-40 mV10 Hz0 pA5 mV1 sec200 pA zap current, pA140 Hzspikes cut200 Hzamplitude of the responseintegratorsresonatorsresonancefrequency of stimulationFigure 7.17: Response of the mesV neuron to injected zap current sweeping through arange of frequencies. Integrators and resonators have different responses.-42integrators-42-42-42-42-42potential (mV)potential (mV)potential (mV)potential (mV)potential (mV)potential (mV)-525 ms-5210 ms-5215 ms0 10 20 30time (ms)0 10 20 30time (ms)0 10 20 30time (ms)resonators-525 ms-5210 ms-5215 ms0 10 20 30time (ms)0 10 20 30time (ms)0 10 20 30time (ms)Figure 7.18: Responses of integrators (top) and resonators (bottom) to input pulseshaving various inter-pulse periods.
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Eugene M. IzhikevichThe Neuroscienc
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6.3.6 Saddle-node homoclinic orbit
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9.4.2 Integrators vs. Resonators .
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xPrefaceversely, cells having quite
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2 Introductionapical dendritessomar
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4 Introduction(a)spikes(b)spikes cu
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6 Introduction1.1.3 Why are neurons
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8 Introductionconcepts of dynamical
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10 Introductionspikingmoderesting m
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12 Introduction(a)spikinglimitcycle
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14 Introductionco-existence of rest
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16 Introduction1.2.4 Neuro-computat
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18 IntroductionspikeFigure 1.16: Ph
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20 Introductionwith coupled neurons
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22 IntroductionFigure 1.17: John Ri
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24 Introduction
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26 Electrophysiology of NeuronsInsi
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28 Electrophysiology of Neuronsouts
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30 Electrophysiology of NeuronsRest
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32 Electrophysiology of NeuronsFigu
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34 Electrophysiology of Neuronsm (V
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36 Electrophysiology of Neurons10.8
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38 Electrophysiology of Neurons2.3
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42 Electrophysiology of NeuronsDepo
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44 Electrophysiology of NeuronsV(x,
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46 Electrophysiology of Neurons1m (
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48 Electrophysiology of NeuronsPara
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52 Electrophysiology of Neuronsvolt
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54 Electrophysiology of Neurons
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56 One-Dimensional SystemsActivatio
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58 One-Dimensional Systemsinwardcur
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60 One-Dimensional Systems40bistabi
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62 One-Dimensional Systemsat every
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64 One-Dimensional Systems?F(V)>0+
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66 One-Dimensional SystemsUnstableE
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70 One-Dimensional Systems+ - - + +
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72 One-Dimensional Systemsλ(V-Veq)
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74 One-Dimensional Systemsbifurcati
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76 One-Dimensional Systems100806040
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78 One-Dimensional Systemsbifurcati
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80 One-Dimensional Systems20 mV100
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82 One-Dimensional Systemsmembrane
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84 One-Dimensional Systemssteady-st
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86 One-Dimensional Systemsspike rig
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88 One-Dimensional SystemsVF (V)1VV
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90 One-Dimensional Systems10.80.60.
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92 One-Dimensional Systems
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94 Two-Dimensional SystemsI Na,pneu
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96 Two-Dimensional Systemsdefines a
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106 Two-Dimensional SystemsFor exam
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120 Two-Dimensional Systemseigenval
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122 Two-Dimensional SystemsK + acti
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124 Two-Dimensional Systemsexcitati
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126 Two-Dimensional SystemsReview o
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128 Two-Dimensional SystemsabcdFigu
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130 Two-Dimensional SystemsFigure 4
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132 Two-Dimensional Systems
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134 Conductance-Based Models• If
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136 Conductance-Based Modelsinward(
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138 Conductance-Based Models1I=01I=
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140 Conductance-Based Modelsleak cu
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144 Conductance-Based Models0I=0ina
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148 Conductance-Based Models1I=651I
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150 Conductance-Based Modelshave co
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152 Conductance-Based Models1 sec10
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154 Conductance-Based Modelsvoltage
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158 Conductance-Based Models5.2.2 E
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160 Conductance-Based Models1recove
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162 Conductance-Based Modelscurrent
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164 Conductance-Based Modelsmodels
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166 Conductance-Based Models
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168 Bifurcationssaddle-node bifurca
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170 Bifurcationssaddle-node bifurca
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172 Bifurcations0.5v 2V-nullclineK
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174 BifurcationsBoth types of the b
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176 BifurcationsV-nullcline0.5K + g
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178 Bifurcationsrstable limit cycle
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180 BifurcationsnVstable limit cycl
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¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
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186 Bifurcationsmembrane potential,
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188 BifurcationsBifurcation of a li
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190 Bifurcationsstableunstablelimit
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Page 236 and 237: 226 Excitabilityspike?spikerestrest
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Page 242 and 243: 232 ExcitabilityClass 3 excitable n
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Page 246 and 247: 236 Excitabilitysaddle-node bifurca
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Page 280 and 281: 270 Excitability50 mV300 msFigure 7
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336 Simple Modelsspiny neuronlatenc
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340 Simple Modelsrat's mitral cell
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342 Bursting(a) cortical chattering
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348 Burstingslow dynamicsneuronvolt
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352 Bursting9.2.1 Fast-slow burster
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356 Bursting0maxmembrane potential,
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362 Burstingmembrane potential, V (
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366 Burstingbifurcation of spiking
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372 Bursting(a)membrane potential,
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spiking380 Burstingfoldbifurcations
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402 Bursting28. [Ph.D.] Develop an
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404 Synchronization (see www.izhike
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408 Solutions to Exercises, Chap. 3
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442 Referencesterneurons mediated b
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444 ReferencesDickson C.T., Magistr
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446 ReferencesGuckenheimer J. (1975
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448 Referencestional Journal of Bif
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450 ReferencesMarkram H, Toledo-Rod
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452 ReferencesRosenblum M.G. and Pi
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454 ReferencesTuckwell H.C. (1988)
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