Inhibitory Neurons and Cortical Circuit Dynamics, Author Summary synaptic plasticity 31 32 and study its responses to external. The brain consists of circuits of neurons that signal to one. another via synapses There are two classes of neurons Results. excitatory cells which cause other neurons to become. more active and inhibitory neurons which cause other Model Description. neurons to become less active It is thought that the Circuit architecture The architecture of the full RS LTS. activity of excitatory neurons is kept in check largely by FS cortical network based on 9 10 is shown in Figure 1 RS. inhibitory neurons when such an inhibitory brake fails a neurons excite RS LTS and FS neurons FS neurons inhibit RS. seizure can result Inhibitory neurons of the low threshold LTS and FS neurons LTS neurons inhibit RS and FS neurons. spiking LTS subtype can potentially fulfill this braking or but not LTS neurons In this article we focus on the short term. anticonvulsant role because the synaptic input to these plasticity of chemical synapses between cortical neurons and. neurons facilitates i e those neurons are active when therefore assume three simplifications First we use firing rate. excitatory neurons are strongly active Using a computa models and effectively average over the spiking dynamics of. tional model we show that because the synaptic output of neurons 29 31 33 34 Second we do not consider electrical. LTS neurons onto excitatory neurons depresses decreases. synapses between cortical interneurons 1 10 Third we assume. with activity the ability of LTS neurons to prevent strong. constant or step external input and do not take into account. cortical activity and seizures is not qualitatively larger than. that of inhibitory neurons of another subtype the fast depression or facilitation of thalamocortical synapses 35. spiking FS cells Furthermore short term one second RS and FS neurons 15 36 but not LTS neurons in layer 4 9. changes in the strength of synapses to and from LTS receive external thalamic input Whether LTS neurons in other. interneurons allow them to shape the behavior of cortical layers are innervated by the thalamus still remains unresolved see. circuits even at modest rates of activity and an RS LTS FS 15 vs 16 17 Therefore we initially study a model in which. circuit is capable of producing slow oscillations on the LTS neurons do not receive thalamic input and analyze the. time scale of these short term changes effects of thalamic input onto LTS neurons separately In addition. LTS neurons are activated by various neuromodulators 7 This. effect is modeled as a reduction of the LTS threshold. varies with the activity of neurons that are presynaptic to cortical We examine the model in four stages First we consider a. neurons e g thalamic relay cells maintaining this balance is a network of RS and LTS neurons where RS neurons receive. dynamic process in which LTS neurons may play an important external inputs either step or absence seizure like Second we. role For example when the firing rate of excitatory neurons is study an RS FS network to demonstrate the differences between. high facilitating excitatory input could generate a supralinear the effects of the FS and LTS populations on the circuit Third we. response of LTS neurons and thus prevent overactivation of consider a full network composed of RS LTS and FS neuronal. excitatory neurons i e activation beyond what is normal leading populations Finally we analyze a slow oscillation state emerging. to pathological behavior This could protect the cortical network from this network. against seizures Consistent with the idea that LTS cells serve a Synaptic dynamics and neuronal firing rates Our. protective function is the observation that selective loss of technical approach makes use of the formulation of Shriki et al. somatostatin positive dendritic targeting interneurons cells similar for rate equations 28 29 30 Each neuronal population is. to neocortical LTS neurons in hippocampus correlates with described by its firing rate M with a subscript i denoting the. epileptic states 21 22 More recently it was suggested that LTS population R for RS F for FS and L for LTS A synaptic. neurons balance excitation and prevent runaway cortical activity connection from a neuron from population j to a neuron from. by decreasing the gain of pyramidal cell output 23 population i is characterized by three dynamic variables with the. subscripts ij the fraction of open synaptic channels s the running. The ability of LTS neurons to protect against network over. fraction of vesicles available for release x and the running value of. activation may be limited however by the depressive nature of. the utilization parameter u 31 32 The variable u quantifies the. LTS to RS inhibitory synapses Furthermore short term synaptic. conditional probability of release of a vesicle in response to an. plasticity can lead to firing patterns more complex than stable. firing rates The existence of two time scales in the system. dynamics the fast time scale of the AMPA receptor and. GABAA receptor mediated postsynaptic potentials PSPs and the. slow time scale of synaptic depression and facilitation processes. may in principle lead to various types of network oscillations or. more complicated patterns Such network oscillations were. observed in previous models of excitatory and inhibitory neurons. 24 25 26 27 but those models did not take into account the. specific physiological characteristics of LTS neurons. In this study we ask by which mechanisms and at what firing. rates do LTS neurons control the activity of cortical circuits. responding to thalamic input and how is control by LTS neurons. different from that of FS neurons To be more specific we. compare the dynamical behavior of LTS neurons with those of FS. neurons in networks with only one type of inhibitory interneuron. Figure 1 Schematic architecture of the RS LTS FS cortical. and in networks with both inhibitory populations to address the circuit Open triangles denote excitatory synapses and solid ellipses. hypothesis of Beierlein et al 9 Silberberg and Markram 18 and denote inhibitory synapses Black lines denote depressing synapses. Kapfer et al 19 We consider a rate model of cortical networks and grey lines denote facilitating synapses. 28 29 30 that includes RS LTS and FS neurons with short term doi 10 1371 journal pcbi 1002248 g001. PLoS Computational Biology www ploscompbiol org 2 October 2011 Volume 7 Issue 10 e1002248. Inhibitory Neurons and Cortical Circuit Dynamics, action potential arriving to the presynaptic terminal assuming that Table 1 Reference parameters for the neuronal populations. vesicle is ready for release before the spike arrives Each synaptic based on 7. connection is characterized by a set of five parameters the efficacy. g the initial conditional probability of release U assuming that a. previous presynaptic spike has not occurred for a long time the Neuronal population h nA b ms21 nA21. decay time of the post synaptic current ts and the recovery time. RS 0 1 0 11, constants from facilitation and depression tf and tr respectively. The dynamics for each synaptic connection are therefore LTS 0 05 0 32. described by the following equations FS 0 28 0 35, doi 10 1371 journal pcbi 1002248 t001. zuij xij Mj 1, RS LTS Networks without RS to RS Recurrent. Connections, We consider a network of two populations composed of RS and. dxij 1 xij, uij xij Mj 2 LTS neurons To explore the role of RS to LTS and LTS to RS. dt tr ij synapses our first step is to study a model with these synaptic. connections only and the effect of the RS to RS synapses will be. studied later RS to LTS synapses facilitate tf LR 670 ms and. duij Uij uij LTS to RS synapses depress tr RL 1250 ms 18 see Methods. zUij 1 uij Mj 3, dt tf ij and Table 2 Therefore xLR 1 uRL URL and equations 1 3. for the RS LTS system become, The firing rates Mi for the three neuronal populations are. determined according to the circuit diagram Figure 1. MR bR IR t zgRR sRR gRL sRL gRF sRF hR z 4, duLR ULR uLR. ML bL gLR sLR gLF sLF hL z 5 zULR 1 uLR MR 8, MF bF IF t zgFR sFR gFL sFL gFF sFF hF z 6 dsRL sRL. zURL xRL ML 9, where for each population Ii t is the external input from sources. outside of the local cortical network hi is the neuronal threshold. dxRL 1 xRL, and bi is the neuronal gain calculated according to the f I curve at URL xRL ML 10. steady state 8 9 The coefficients of synaptic conductances are dt tr RL. denoted by gij and the total synaptic input from neuronal. population j to a neuron from population i is gij sij The function Steady state firing When the input to the RS population. is the rectification linear threshold function x x for x 0 IR is constant in time the steady state values of the system are. and x 0 otherwise Note that the currents Ii and the. conductances gij are measured in arbitrary units 30. Model parameters Despite the fact that our model is. relatively simple it includes many parameters Therefore it is. important to consider ranges of biophysical parameters It is of. course impossible to study the entire multidimensional space of Table 2 Reference parameters for the synapses between the. parameters We limit the range of parameters by taking most of various types of neurons. their values from the literature but some of them remain. unknown In particular the maximal synaptic conductances a. neuron receives from its presynaptic neurons are often hard to Synaptic. determine Knowing these difficulties we use the following connection ts ms tf ms tr ms U g Reference. strategy that we have often used in the past e g 37 We RSrRS 2 0 463 0 21 5 83. choose a biophysically plausible parameter set as a reference point RSrLTS 6 3 0 1250 0 3 35 18. in the parameter space The reference parameter values for the. LTSrRS 2 670 0 0 09 7 18, model are written in Tables 1 and 2 see Methods Starting from. RSrFS 2 0 875 0 14 38 47, this point we vary one or two parameters to study their effect. Specifically we study sub networks of RS LTS and RS FS FSrRS 2 0 227 0 3 18 47 84. populations to investigate the respective role of the two types of FSrLTS 2 0 400 0 3 5. interneurons before studying the full RS LTS FS network LTSrFS 2 0 400 0 3 10. Exploring the dependence on parameters provides us with an FSrFS 2 0 400 0 3 20. understanding of the different dynamical patterns the network can. exhibit doi 10 1371 journal pcbi 1002248 t002, PLoS Computational Biology www ploscompbiol org 3 October 2011 Volume 7 Issue 10 e1002248. Inhibitory Neurons and Cortical Circuit Dynamics, ts LR ULR 1ztf LR MR MR weakly with IR Since tr RL URL ML 1 equation 12 becomes. sLR 11 sRL URL ts RL ML and dMR dIR just above IR LTS th is. 1ztf LR ULR MR, Equations 24 25, ts RL URL ML dMR 1 1 ULR. sRL 12 zgRL URL ts RL bL gLR ts LR 1 15, 1ztr RL URL ML dIR bR 1zULR tf LR MR 2. i e the slope dMR dIR at threshold scales like 1 gRL for large gRL. where For large input IR the firing rates MR and ML are large as well. and sRL ts RL tr RL Equation 12 Using equation 13 we obtain. MR bR IR gRL sRL hR z 13, MR bR IR gRL ts RL tr RL hR 16. ML bL gLR sLR hL z 14 Therefore MR increases linearly with IR with a slope gain bR. and is reduced by inhibition by a constant value bR gRL ts RL tr RL. Like MR ML increases linearly with IR for large IR. The firing rates of the two populations MR and ML as functions ML bL gLR ts LR MR hL Figure 2A The gains of MR and. of IR for several values of the LTS to RS synaptic conductance ML with IR remain relatively small in an IR range of about gRL. coefficient gRL are shown in Figure 2A When gRL 0 the RS ts RL tr RL before they reach approximately their maximal values. population is silent for IR hR and MR increases linearly with The reduction of activity by a constant value at large IR and. IR2hR for IR hR LTS neurons fire for IR IR LTS th large firing rates is a result of the properties of the depressing. IR LTS th hR Threshold for LTS firing for gRR 0 in LTS to RS synapses at high firing rates ML The postsynaptic. Methods Equations 22 23 and inhibit RS neurons for gRL 0 current PSC amplitude for such a synapse is inversely. For IR just above IR LTS th ML is small and MR increases only proportional to ML the firing rate of the presynaptic neuron. Figure 2 Steady state response of the RS LTS network with gRR 0 to constant inputs to the RS neurons A MR IR curves top panel. and ML IR curves bottom panel are plotted for gRL 0 black 17 5 red and 35 green Additional parameters are tr RL 1250 ms gLR 7 5 The. arrow below the abscissa in the top panel points to the value of IR LTS th B MR IR curves are plotted for tr RL 1250 ms black 500 ms red and. 150 ms green and 0 blue Additional parameters are gRL 35 gLR 7 5 C MR IR curves are plotted for gLR 0 black 7 5 red 15 green and 22 5. blue Additional parameters are gRL 35 tr RL 1250 ms. doi 10 1371 journal pcbi 1002248 g002, PLoS Computational Biology www ploscompbiol org 4 October 2011 Volume 7 Issue 10 e1002248. Inhibitory Neurons and Cortical Circuit Dynamics, 38 and therefore the total LTS to RS inhibition is independent. of ML This constant inhibition shifts the MR IR curve to the right. by a fixed value and this shift is translated to a constant reduction. of MR because of the linear dependency of MR on the total input. to the neuronal population Indeed the inhibitory effect on the. MR IR curve is enhanced when tr RL is small and LTS to RS. LTS and FS Inhibitory Interneurons Short Term Synaptic Plasticity and Cortical Circuit Dynamics Itai Hayut1 2 Erika E Fanselow3 4 Barry W Connors3 David Golomb1 1Department of Physiology and Neurobiology and Zlotowski Center for Neuroscience Faculty of Health Sciences Ben Gurion University Be er Sheva Israel 2Department

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