Simple Model of Spiking Neurons - Izhikevich

1 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 6, NOVEMBER 20031569 Simple Model of Spiking NeuronsEugene M. IzhikevichAbstract A Model is presented that reproduces Spiking and burstingbehavior of known types of cortical Neurons . The Model combines the bi-ologically plausibility of Hodgkin Huxley-type dynamics and the compu-tational efficiency of integrate-and-fire Neurons . Using this Model , one cansimulate tens of thousands of Spiking cortical Neurons in real time (1 msresolution) using a desktop Terms Bursting, cortex, Hodgkin Huxley, PCNN, quadratic in-tegrate-and-fire, Spiking , INTRODUCTIONTo understand how the brain works, we need to combine experi-mental studies of animal and human nervous systems with numericalsimulation of large-scale brain models. As we develop such large-scalebrain models consisting of Spiking Neurons , we must find compromisesbetween two seemingly mutually exclusive requirements: The modelfor a single neuron must be: 1) computationally Simple , yet 2) capableof producing rich firing patterns exhibited by real biological biophysically accurate Hodgkin Huxley-type models is compu-tationally prohibitive, since we can simulate only a handful of neuronsin real time.

2 In contrast, using an integrate-and-fire Model is computa-tionally effective, but the Model is unrealistically Simple and incapableof producing rich Spiking and bursting dynamics exhibited by this paper, a Simple Spiking Model (1), (2) is presented that is asbiologically plausible as the Hodgkin Huxley Model , yet as computa-tionally efficient as the integrate-and-fire Model . Depending on four pa-rameters, the Model reproduces Spiking and bursting behavior of knowntypes of cortical Neurons , as we illustrate in Fig. 1 and summarize inFig. analysis of the Model will be published in the mono-graph by Izhikevich [8]. The derivation of the first (1) is based on bi-furcation theory and normal form reduction [2], [5], and the partv0=v2+Iis sometimes referred to as being a quadratic integrate-and-fireneuron. The full Model was first published in [10, eqns. (4) and (5)with voltage reset discussed in Sect.]

3 ] in a trigonometric formmore suitable for mathematical analysis. The form presented here ismore suitable for large-scale THEMODELB ifurcation methodologies [8] enable us to reduce many biophysi-cally accurate Hodgkin Huxley-type neuronal models to a two-dimen-sional (2-D) system of ordinary differential equations of the formv0=0:04v2+5v+140 u+I(1)u0=a(bv u)(2)with the auxiliary after-spike resettingifv 30mV;the nv cu u+d:(3)Here,vanduare dimensionless variables, anda,b,c, anddare dimen-sionless parameters, and0=d=dt, wheretis the time. The variableManuscript received February 2003; revised April 2003. This work was sup-ported in part by the Neurosciences Research Foundation and by a Grant fromthe Alafi Family author is with The Neurosciences Institute, San Diego, CA 92121 USA(e-mail: Object Identifier 1. The Simple Model (1), (2) can reproduce firing patters of neuronsrecorded from the rat s motor cortex.)

4 Data are kindly shared by N. Desai, Model parameters as in Fig. the membrane potential of the neuron andurepresents amembrane recovery variable, which accounts for the activation ofK+ionic currents and inactivation ofNa+ionic currents, and it providesnegative feedback tov. After the spike reaches its apex (+30 mV), themembrane voltage and the recovery variable are reset according to the(3). Synaptic currents or injected dc-currents are delivered via the part0:04v2+5v+140was obtained by fitting the spike initiationdynamics of a cortical neuron (other choices also feasible) so that themembrane potentialvhasmVscale and the timethasmsscale. Theresting potential in the Model is between 70 and 60 mV dependingon the value ofb. As most real Neurons , the Model does not have a fixedthreshold; Depending on the history of the membrane potential prior tothe spike, the threshold potential can be as low as 55 mV or as highas 40 mV.

5 The parameteradescribes the time scale of the recovery variableu. Smaller values result in slower recovery. A typical value isa=0:02. The parameterbdescribes the sensitivity of the recovery variableuto the subthreshold fluctuations of the membrane values couplevandumore strongly resulting in possiblesubthreshold oscillations and low-threshold Spiking dynamics. Atypical value isb=0:2. The caseb<a(b> a)correspondsto saddle-node (Andronov Hopf) bifurcation of the resting state[10]. The parametercdescribes the after-spike reset value of the mem-brane potentialvcaused by the fast high-thresholdK+conduc-tances. A typical value isc= $ 2003 IEEE1570 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 6, NOVEMBER 2003 Fig. 2. Known types of Neurons correspond to different values of the parametersa,b,c,din the Model described by the (1), (2). RS, IB, and CH are corticalexcitatory Neurons .

6 FS and LTS are cortical inhibitory interneurons. Each inset shows a voltage response of the Model neuron to a step of dc-currentI=10(bottom). Time resolution is ms. This figure is reproduced with permission from (Electronic version of the figure and reproductionpermissions are freely available at ) The parameterddescribes after-spike reset of the recovery vari-ableucaused by slow high-thresholdNa+andK+ typical value isd= choices of the parameters result in various intrinsic firing pat-terns, including those exhibited by the known types of neocortical [1],[3], [4] and thalamic Neurons as summarized in Fig. 2. A possible exten-sion of the Model (1), (2) is to treatu,aandbas vectors, and useuinstead ofuin the voltage (1). This accounts for slow conductanceswith multiple time scales, but we find such an extension unnecessarilyfor cortical DIFFERENTTYPES OFDYNAMICSN eocortical Neurons in the mammalian brain can be classified intoseveral types according to the pattern of Spiking and bursting seen inintracellular recordings.

7 All excitatory cortical cells are divided into thefollowing four classes [1], [3]: RS (regular Spiking ) Neurons are the most typical Neurons in thecortex. When presented with a prolonged stimulus (injected stepof dc-current in Fig. 2RS, bottom) the Neurons fire a few spikeswith short interspike period and then the period increases. This iscalled the spike frequency adaptation. Increasing the strength ofthe injected dc-current increases the interspike frequency, thoughit never becomes too fast because of large spike-afterhyperpolar-izations. In the Model , this corresponds toc= 65m V(deepvoltage reset) andd=8(large after-spike jump ofu). IB (intrinsically bursting) Neurons fire a stereotypical burst ofspikes followed by repetitive single spikes (Fig. 2IB). In themodel, this corresponds toc= 55mV(high voltage reset)andd=4(large after-spike jump ofu). During the initial burst,variableubuilds up and eventually switches the dynamics frombursting to Spiking .

8 CH (chattering) Neurons can fire stereotypical bursts of closelyspaced spikes. The inter-burst frequency can be as high as 40 the Model , this corresponds toc= 50mV(very high voltagereset) andd=2(moderate after-spike jump ofu).Allinhibitorycortical cells are divided into the following two classes[4]: FS (fast Spiking ) Neurons can fire periodic trains of action poten-tials with extremely high frequency practically without any adap-tation (slowing down), as one can see in Fig. 2FS. In the Model ,this corresponds toa=0:1(fast recovery). LTS (low-threshold Spiking ) Neurons can also fire high-frequencytrains of action potentials (Fig. 2 LTS), but with a noticeable spikefrequency adaptation. These Neurons have low firing thresholds,which is accounted for byb=0:25in the Model . To achieve abetter quantitative fit with real LTS Neurons , other parameters ofthe Model need to be changed as addition, our Model can easily reproduce behavior of thalamo-cor-tical Neurons , which provide the major input to the cortex TC (thalamo-cortical) Neurons have two firing regimes: Whenat rest (vis around 60 mV) and then depolarized, they exhibitIEEE TRANSACTIONS ON NEURAL NETWORKS, VOL.

9 14, NO. 6, NOVEMBER 20031571 Fig. 3. Simulation of a network of 1000 randomly coupled Spiking : spike raster shows episodes of alpha and gamma band rhythms (verticallines). Bottom: typical Spiking activity of an excitatory neuron. All spikes wereequalized at+30 mV by resettingvfirst to+30 mV and then firing as in Fig. 2TC, left voltage trace. However, if a neg-ative current step is delivered so that the membrane potential ishyperpolarized (vis around 90 mV), the Neurons fire a reboundburst of action potentials, as in Fig. 2TC, right voltage Model can exhibit other interesting types of dynamics. RZ (resonator) Neurons have damped or sustained subthresholdoscillations, as in Fig. 2RZ. They resonate to rhythmic inputshaving appropriate frequency (as the resonate-and-fire Model [9]). This behavior corresponds toa=0:1andb=0 that there is a bistability of resting and repetitive spikingstates: The neuron can be switched between the states by anappropriately timed brief of other neuronal types, including those in brainstem, hip-pocampus, basal ganglia, and olfactory bulb, can also be described byour one-fits-all choice of the function0:04v2+5v+140in (1) isjustified when large-scale networks of Spiking Neurons are simulated,as we discuss below.

10 However, if one is interested in the behavior ofa single neuron, then other choices of the function are available, andsometimes more preferable. For example, the function0:04v2+4:1v+108withb= 0:1is a better choice for the RS neuron, since it leads tothe saddle-node on invariant circle bifurcation and Class 1 excitability[10].IV. PULSE-COUPLEDIMPLEMENTATIONWe have used this Model to simulate a sparse network of 10 000spiking cortical Neurons with 1 000 000 synaptic connections in realtime (resolution 1 ms) using a 1 GHz desktop PC and C++ program-ming language. The following MATLAB program (also available onauthor s webpage) simulates a network of randomly connected 1000neurons in real time. Motivated by the anatomy of a mammalian cortex,we choose the ratio of excitatory to inhibitory Neurons to be 4 to 1, andwe make inhibitory synaptic connections stronger.