In this article under material and methods I need you to recreate section 2.1 only in matlab and do not worry about I_syn. I need the matlab code and proof that it worked on your end. You can use this code attached as a skeleton.
Gamma Oscillation by Synaptic Inhibition in a Hippocampal Interneuronal Network Model Xiao-Jing Wang1 and György Buzsáki2 1Physics Department and Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02254, and 2Center for Molecular and Behavioral Neuroscience, Rutgers University, Newark, New Jersey 07102 Fast neuronal oscillations (gamma, 20–80 Hz) have been observed in the neocortex and hippocampus during behav- ioral arousal. Using computer simulations, we investigated the hypothesis that such rhythmic activity can emerge in a random network of interconnected GABAergic fast-spiking interneurons. Specific conditions for the population synchro- nization, on properties of single cells and the circuit, were identified. These include the following: (1) that the amplitude of spike afterhyperpolarization be above the GABAA synaptic reversal potential; (2) that the ratio between the synaptic decay time constant and the oscillation period be sufficiently large; (3) that the effects of heterogeneities be modest be- cause of a steep frequency–current relationship of fast- spiking neurons. Furthermore, using a population coherence measure, based on coincident firings of neural pairs, it is demonstrated that large-scale network synchronization re- quires a critical (minimal) average number of synaptic con- tacts per cell, which is not sensitive to the network size. By changing the GABAA synaptic maximal conductance, synaptic decay time constant, or the mean external excitatory drive to the network, the neuronal firing frequencies were grad- ually and monotonically varied. By contrast, the network syn- chronization was found to be high only within a frequency band coinciding with the gamma (20–80 Hz) range. We conclude that the GABAA synaptic transmission provides a suitable mecha- nism for synchronized gamma oscillations in a sparsely con- nected network of fast-spiking interneurons. In turn, the inter- neuronal network can presumably maintain subthreshold oscillations in principal cell populations and serve to synchro- nize discharges of spatially distributed neurons. Key words: gamma rhythm; hippocampus; interneurons; GABAA; synchronization; computer model Although fast gamma cortical oscillation has been the subject of active investigation in recent years (cf. Singer and Gray, 1995), its underlying neuronal mechanisms remain elusive. Two major is- sues are the cellular origin of rhythmicity (Llinás et al., 1991; McCormick et al., 1993; Wang, 1993) and the mechanism(s) of large-scale population synchronicity (Freeman, 1975; Bush and Douglas 1991; Engel et al., 1991; Hansel and Sompolinsky, 1996). Traditionally, recurrent excitation between principal (pyramidal) neurons is viewed as a major source of rhythmogenesis as well as neuronal synchronization. However, in model studies in which quantitative data about the synaptic time course were incorpo- rated, it was found that glutamatergic synaptic excitation of the AMPA type usually desynchronizes rather than synchronizes re- petitive spike firings of mutually coupled neurons (Hansel et al., 1995; van Vreeswijk et al., 1995). Therefore, recurrent connec- tions between pyramidal cells alone do not seem to account for the network coherence during cortical gamma oscillations. It was suggested that pyramidal cell populations may be entrained by synchronous rhythmic inhibition originating from fast-spiking in- terneurons (Buzsáki et al., 1983; Lytton and Sejnowski, 1991). During field gamma oscillations, intracellular recordings from pyramidal cells revealed both EPSPs and IPSPs phase-locked to the field oscillation frequencies (Jagadeesh et al., 1992; Chen and Fetz, 1993; Soltész and Deschênes, 1993). In this paper, we address the question whether, in the hip- pocampus, an interneuronal network can generate a coherent oscillatory output to the pyramidal neurons, thereby providing a substrate for the synaptic organization of coherent gamma popu- lation oscillations. In the behaving rat, physiologically identified interneurons were shown to fire spikes in the gamma frequency range and phase-locked to the local field waves (Bragin et al., 1995). Intracellular studies and immunochemical staining demon- strated that these interneurons are interconnected via GABAergic synapses (Lacaille et al., 1987; Sik et al., 1995; Gulyás et al., 1996). Theoretical studies suggest that these GABAergic interconnec- tions may synchronize an interneuronal network when appropri- ate conditions on the time course of synaptic transmission are satisfied (Wang and Rinzel, 1992, 1993; van Vreeswijk et al., 1995). Moreover, in a recent in vitro experiment (Whittington et al., 1995; Traub et al., 1996), the excitatory glutamate AMPA and NMDA synaptic transmissions were blocked in the hippocampal slice. When metabotropic glutamate receptors were activated, transient oscillatory IPSPs in the 40 Hz frequency range were observed in pyramidal cells. These IPSPs were assumed to origi- nate from the firing activities of fast-spiking interneurons synchro- nized by their interconnections. Computer simulations (Whitting- ton et al., 1995; Traub et al., 1996) lend further support to this hypothesis. To assess whether an interneuronal network can subserve an adequate basis for the gamma frequency population rhythm in the hippocampus, it is necessary to identify its specific requirements Received May 5, 1996; revised June 25, 1996; accepted July 31, 1996. This work was supported by the National Institute of Mental Health (MH53717- 01), Office of Naval Research (N00014-95-1-0319), and the Sloan Foundation to X.J.W.; and HFSP and the National Institute of Neurological Disease and Stroke (NS34994) to G.B. and X.J.W. Simulations were partly performed at the Pittsburgh Supercomputing Center. We thank D. Golomb, D. Hansel, J.-C. Lacaille, and C. McBain for discussions, A. Sik for preparing Figure 2, and L. Abbott, J. Lisman, and R. Traub for carefully reading this manuscript. Correspondence should be addressed to Xiao-Jing Wang, Center for Complex Systems, Brandeis University, Waltham, MA 02254. Copyright q 1996 Society for Neuroscience 0270-6474/96/166402-12$05.00/0 The Journal of Neuroscience, October 15, 1996, 16(20):6402–6413 on the cellular properties and network connectivities, as well as to determine whether these conditions are satisfied by particular interneuronal subtypes. The present study is devoted to investi- gate such requirements using computer simulations. We found that synaptic transmission via GABAA receptors in a sparsely connected network of model interneurons can provide a mecha- nism for gamma frequency oscillations, and we compared the modeling results with the anatomical and electrophysiological data from hippocampal fast spiking interneurons. MATERIALS AND METHODS Model neuron. Each interneuron is described by a single compartment and obeys the current balance equation: Cm dV dt 5 2INa 2 IK 2 IL 2 Isyn 1 Iapp , (2.1) where Cm 5 1 mF/cm 2 and Iapp is the injected current (in mA/cm 2). The leak current IL 5 gL(V 2 EL) has a conductance gL 5 0.1 mS/cm 2, so that the passive time constant t0 5 Cm/gL 5 10 msec; EL 5 265 mV. The spike-generating Na1 and K1 voltage-dependent ion currents (INa and IK) are of the Hodgkin–Huxley type (Hodgkin and Huxley, 1952). The transient sodium current INa 5 gNam` 3 h(V 2 ENa), where the acti- vation variable m is assumed fast and substituted by its steady-state function m` 5 am/(am 1 bm); am(V ) 5 20.1(V 1 35)/(exp(20.1(V 1 35)) 2 1), bm(V ) 5 4exp(2(V 1 60)/18). The inactivation variable h obeys a first-order kinetics: dh dt 5 f~ah~1 2 h! 2 bhh! (2.2) where ah(V ) 5 0.07 exp(2(V 1 58)/20) and bh(V ) 5 1/(exp(20.1(V 1 28)) 1 1). gNa 5 35 mS/cm 2; ENa 5 55 mV, f 5 5. The delayed rectifier IK 5 gKn 4 (V 2 EK), where the activation variable n obeys the following equation: dn dt 5 f~an~1 2 n! 2 bnn! (2.3) with an(V )5 20.01(V1 34)/(exp(20.1(V1 34))2 1) and bn(V )5 0.125 exp(2(V 1 44)/80); gK 5 9 mS/cm 2, and EK 5 290 mV. These kinetics and maximal conductances are modified from Hodgkin and Huxley (1952), so that our neuron model displays two salient prop- erties of hippocampal and neocortical fast-spiking interneurons. First, the action potential in these cells is followed by a brief afterhyperpolarization (AHP) of about 215 mV measured from the spike threshold of approx- imately 255 mV (McCormick et al., 1985; Lacaille and Williams, 1990; Morin et al., 1995; Zhang and McBain, 1995). Thus, during the spike repolarization the membrane potential reaches a minimum of about 270 mV, rather than being close to the reversal potential of the K1 current, EK 5 290 mV. This is accomplished in the model by relatively small maximal conductance gK and fast gating process of IK so that it deacti- vates quickly during spike repolarization. Second, these interneurons have the ability to fire repetitive spikes at high frequencies (with the frequency–current slope up to 200–400 Hz/ nA) (McCormick et al., 1985; Lacaille and Williams, 1990; Zhang and McBain, 1995). With fast kinetics of the inactivation (h) of INa, the activation (n) of IK, and the relatively high threshold of IK, the model interneuron displays a large range of repetitive spiking frequencies in response to a constant injected current Iapp (Fig. 1A, left). It has a small current threshold (the rheobase Iapp . 0.2 mA/cm2), and the firing rate is as high as 400 Hz for Iapp . 20 mA/cm2. Similar to cortical interneurons (McCormick et al., 1985; Lacaille and Williams, 1990), the whole fre- quency–current curve is not linear, and the frequency–current slope is larger at smaller Iapp values (lower frequencies) (Fig. 1A, right). As a consequence, the neural population is more sensitive to input heteroge- neities at smaller Iapp values. This is demonstrated in Figure 1B, where a Gaussian distribution of Iapp is applied to a population of uncoupled neurons (N 5 100), with a mean Im and standard deviation Is. Given a fixed and small Is 5 0.03, the mean drive Im is varied systematically, and the resulting dispersion in the neuronal firing frequencies, fs /fm (standard deviation of the firing rate/mean firing rate) is shown as function of Im (Fig. 1B, top). When plotted versus fm, it is evident that with the same amount of dispersion in applied current (Is) the dispersion in firing rates fs /fm is dramatically increased for fm , 20 Hz (Fig. 1B, bottom). This feature has important implications for the frequency-dependent network behaviors (see Results). Model synapse. The synaptic current Isyn 5 gsyns(V 2 Esyn), where gsyn is the maximal synaptic conductance and Esyn is the reversal potential. Typically, we set gsyn 5 0.1 mS/cm 2 and Esyn 5 275 mV (Buhl et al., 1995). The gating variable s represents the fraction of open synaptic ion channels. We assume that s obeys a first-order kinetics (Perkel et al., 1981; Wang and Rinzel 1993): ds dt 5 aF~Vpre!~1 2 s! 2 bs, (2.4) where the normalized concentration of the postsynaptic transmitter- receptor complex, F(Vpre), is assumed to be an instantaneous and sigmoid function of the presynaptic membrane potential, F(Vpre) 5 1/(1 1 exp(2(Vpre 2 usyn)/2)), where usyn (set to 0 mV) is high enough so that the transmitter release occurs only when the presynaptic cell emits a spike. Figure 1. Model of single neuron and synapse. A, Left, Firing frequency versus applied current intensity ( f 2 Iapp curve) of the model neuron. The firing rate can be as high as 400 Hz. Right, The derivative df/dIapp shows that the f/Iapp slope is much larger at smaller Iapp (lower f ) values. B, Dispersion in firing rates caused by heterogeneity in input current. A Gaussian distribution for input currents, with standard deviation Is 5 0.03, is applied to a population of uncoupled neurons. The dispersion in firing rates was computed as the ratio between the standard deviation and the mean of firing rates ( fs /fm). This ratio is much larger for smaller mean current amplitude Im (top). Plotting fs /fm versus fm shows that the disper- sion in firing rates is dramatically increased for fm , 20 Hz (bottom). C, A brief current pulse applied to a presynaptic cell generates a single action potential, which elicits an inhibitory postsynaptic current (Isyn) and membrane potential change in a postsynaptic cell (gsyn 5 0.1 mS/cm 2). Wang and Buzsáki • Gamma Rhythm in an Interneuronal Network J. Neurosci., October 15, 1996, 16(20):6402–6413 6403 The channel opening rate a 5 12 msec21 assures a fast rise of the Isyn, and the channel closing rate b is the inverse of the decay time constant of the Isyn; typically, we set b 5 0.1 msec 21 (tsyn 5 10 msec). An example of Isyn and IPSP elicited by a single presynaptic spike is illustrated in Figure 1C. Random network connectivity. The network model consists of N cells. The coupling between neurons is randomly assigned, with a fixed average number of synaptic inputs per neuron,Msyn. The probability that a pair of neurons are connected in either direction is p 5 Msyn/N. For comparison, we also used fully coupled (all-to-all) connectivity (Msyn 5 N ). In the model, the maximal synaptic conductance gsyn is divided by Msyn, so that when the number of synapses Msyn is varied, the total synaptic drive per cell in average remains the same. Msyn is the convergence/divergence factor of the neural coupling in the network. Experimentally, an estimate of this important quantity has been obtained for an interneuronal network of the CA1 hippocampus (Sik et al., 1995). A parvalbumin-positive (PV1) basket interneuron was stained intracellularly by biocytin in vivo. Its axonal arborization was largely confined in the striatum pyramidale (Fig. 2A). Other PV1 interneurons were stained immunochemically, and the contacts made by the biocytin- filled cell with other PV1 cells were counted (Sik et al., 1995). It was concluded that a single PV1 basket cell makes synaptic contacts with at least 60