Using computational cortical-network models, the authors show that partial NMDA-receptor blockade that preferentially affects inhibitory interneurons paradoxically increases gamma oscillations and overall network responsiveness. This hyperexcitable state provides a mechanistic explanation for ketamine- and schizophrenia-associated increases in gamma power and exaggerated responses to sensory input, consistent with hallucinations.
Psychotic drugs such as ketamine induce symptoms close to schizophrenia, and stimulate the production of gamma oscillations, as also seen in patients, but the underlying mechanisms are still unclear. Here, we have used computational models of cortical networks generating gamma oscillations, and have integrated the action of drugs such as ketamine to partially block n-methyl-d-Aspartate (NMDA) receptors. The model can reproduce the paradoxical increase of gamma oscillations by NMDA-receptor antagonists, assuming that antagonists affect NMDA receptors with higher affinity on inhibitory interneurons. We next used the model to compare the responsiveness of the network to external stimuli, and found that when NMDA channnels are diminished, an increase of gamma power is observed altogether with an increase of network responsiveness. However, this responsiveness increase applies not only to gamma states, but was also present in asynchronous states with no apparent gamma. We conclude that NMDA antagonists induce an increased excitability state, which may or may not produce gamma oscillations, but the response to external inputs is exacerbated, which may explain phenomena such as altered perception or hallucinations. Significance Statement n-methyl-d-Aspartate (NMDA) synaptic receptors mediate excitatory interactions using the neurotransmitter glutamate. NMDA receptors have been implicated in psychosis such as schizophrenia and are also targeted by hallucinogenic drugs like Ketamine. However, the exact mechanisms of action are sill unclear. Furthermore, Ketamine paradoxially leads to and excited state, while it is a blocker of NMDA receptors, therefore in principle diminishing excitation. Here, we use models of cortical networks generating gamma oscillations, and show that this model can explain the paradoxical exciting effect of Ketamine if one assumes a higher affinity on NMDA receptors of inhibitory interneurons. The simulated Ketamine effect reproduces known symptoms of psychosis such as increased gamma oscillations and exacerbated responses to external inputs, compatible with hallucinations.
Schizophrenia is described as a disorder with positive, negative and cognitive symptoms, and previous work has identified alterations in neurotransmitter systems, anatomy and neural rhythms in affected patients. In particular, many studies report increased power and/or phase synchrony of gamma-band oscillations in early-course patients, and higher gamma activity correlates with greater psychotic symptom load. Sub‑anaesthetic doses of NMDA receptor (NMDAR) antagonists such as ketamine produce a transient psychotic state in humans and animals that resembles schizophrenia and also boost gamma power, but the circuit mechanisms linking NMDAR hypofunction, increased gamma and altered perception remain uncertain. Susin and colleagues set out to explore how partial blockade of NMDARs alters cortical network dynamics and responsiveness, using a biophysically informed computational network model. The study aims to reproduce experimental features induced by NMDAR antagonists, test the hypothesis that preferential blockade of NMDARs on inhibitory interneurons can account for paradoxical network excitation, and to examine whether the enhanced gamma accompanying NMDAR block is responsible for changes in network responsiveness to external inputs.
The investigators implemented a spiking network model composed of two cell types: 4,000 Regular Spiking (RS) excitatory neurons and 1,000 Fast Spiking (FS) inhibitory neurons (total N = 5,000). Neurons were modelled with the Adaptive Exponential Integrate-and-Fire (AdEx) formalism, which includes spike-triggered and subthreshold adaptation terms. Connectivity was random with a connection probability of 10%, yielding on average 500 excitatory and 100 inhibitory presynaptic inputs per neuron. Synaptic currents comprised AMPA and NMDA excitatory components and GABAA inhibitory currents. AMPA and GABAA conductances were implemented as fast exponential decays (τAMPA decay = 1.5 ms, τGABAA decay = 7.5 ms), while NMDA conductances used a biexponential kernel (τNMDA rise = 2 ms, τNMDA decay = 200 ms) and included a voltage-dependent magnesium block term. Synaptic strengths for AMPA and GABAA were set to values taken from prior work (QAMPA = 5 nS, QGABAA = 3.34 nS) and all synapses had a 1.5 ms delay. The NMDA/AMPA charge ratio was on average larger in RS than in FS cells, consistent with experimental reports. External drive to mimic cortical input consisted of Next = 5,000 independent excitatory Poisson spike trains connected with 10% probability and QAMPA Ext = 0.8 nS. Two baseline drive regimes were used: µext = 3 Hz to elicit gamma activity and µext = 2 Hz to produce asynchronous‑irregular (AI) states. To probe responsiveness, the authors added a time‑varying Gaussian-shaped increase in the external Poisson rate (standard deviation 50 ms) of variable amplitude, applied to both AMPA and NMDA external inputs. NMDAR antagonism was modelled by reducing the NMDA synaptic weights QNMDA onto RS and FS populations along trajectories in the (QNMDA RS, QNMDA FS) parameter space intended to mimic progressive channel block. Simulations were run in the Brian2 simulator with Euler integration (dt = 0.1 ms). Population activity was summarised via a simulated local field potential (LFP) obtained by convolving spike trains with an experimentally derived unitary LFP kernel; for spectral analyses the Welch method was used (Hamming window 0.25 s, 125 overlapping points). Responsiveness R was defined as the spike-count difference between stimulated and unstimulated conditions, normalised by network size and time window.
Simulations reproduced several empirical features associated with sub‑anaesthetic NMDAR antagonists. As NMDA synaptic strengths were decreased, the model showed an increase in the firing rate of RS excitatory neurons and a decrease in firing of FS inhibitory neurons. Concomitantly, the population-level gamma-band power increased under partial NMDA blockade. The authors obtained these effects when the NMDAR block affected interneurons preferentially; parameter sets with larger decreases of QNMDA onto FS cells produced network excitation and boosted gamma power that match experimental observations. The study distinguished two dynamical quantities: excitability (overall spontaneous spiking) and responsiveness (additional spikes elicited by a transient external stimulus). Across a range of stimulus amplitudes, responsiveness of RS cells increased as NMDA blockade deepened, whereas responsiveness of FS cells decreased. Thus, in the model both excitability and RS responsiveness tended to increase together under interneuron‑biased NMDAR block. Comparisons between gamma states and asynchronous‑irregular (AI) states showed that AI states consistently yielded higher responsiveness than gamma states for the same stimulus protocol. Importantly, the increase in responsiveness associated with NMDAR antagonism was not specific to gamma oscillatory regimes: enhanced responsiveness was observed both during pronounced gamma rhythms and during AI regimes lacking apparent gamma. Mechanistically, the authors attribute the responsiveness change to oppositely directed membrane shifts: RS cells became depolarised under NMDAR antagonism while FS cells were relatively hyperpolarised, producing greater stimulus‑evoked output from excitatory cells. The model also reports that the NMDA/AMPA charge balance is higher in RS neurons than in FS neurons, a feature used in constructing the parameter sets.
Susin and colleagues interpret their results as supporting a parsimonious circuit mechanism linking NMDAR hypofunction to increased gamma and to abnormal sensory responses. They highlight three principal findings: (1) modulation of gamma rhythms by partial NMDAR block in a two‑population PING (pyramidal‑interneuron gamma) network is reproduced when antagonism predominantly affects interneuronal NMDARs; (2) partial NMDAR blockade is accompanied by increased network responsiveness to external inputs; and (3) the responsiveness increase is not contingent on the presence of gamma oscillations, because asynchronous states showed similar or even greater responsiveness. The authors place these findings in the context of experimental literature that reports interneuron vulnerability to NMDAR antagonists and ketamine‑induced disinhibition of glutamatergic neurons. Their model reproduces experimental observations—including increased glutamatergic activity and enhanced gamma power—only when the NMDA synaptic strengths on FS interneurons are decreased more than on RS cells, which they argue supports the hypothesis that hypofunction in parvalbumin‑positive fast spiking interneurons is important for schizophrenia‑like phenotypes. At the same time the paper acknowledges contrasting experimental reports that question the relative impact of NMDAR block on excitatory versus inhibitory neurons, and notes that the precise cellular locus of NMDAR hypofunction in schizophrenia remains unresolved. Regarding clinical and mechanistic implications, the investigators propose that the increased responsiveness of cortical networks under NMDAR antagonism could underlie exacerbated sensory responses and phenomena such as altered perception or hallucinations observed with ketamine and in psychosis. They suggest empirical tests of the model predictions, for instance comparing stimulus‑evoked responses in cortical slices or in vivo before and after application of NMDAR antagonists. The authors are careful to report that some experimental results are conflicting and that their model assumptions—most notably the interneuron‑biased NMDAR block—are critical for reproducing the observed phenomena.
S CHIZOPHRENIA is a mental disorder characterized by three classes of symptoms: positive symptoms (such as delusions, hallucinations and disordered thoughts or speech), negative symptoms (comprehending poverty of speech and deficits of normal emotional response), and cognitive deficits. Several abnormalities have been identified in schizophrenic patients, including important differences in neurotransmitters systems, anatomical deficits and abnormal neural rhythms. Gamma oscillations in early-course schizophrenia patients are commonly reported to present increased power and/or phase synchronization. In parallel, positive correlation between psychotic symptoms and the Gamma power have been identified in schizophrenic patients, in which higher Gamma-band activity corresponded to increased symptom load. These findings indicate that hallucinations and delusions could be related to an excess of oscillatory synchronization in the Gamma band. NMDA receptor (NMDAR) antagonists, commonly used in sub-anesthetic doses as animal and human models to study Schizophrenia, induce a psychotic state that resembles all three classes of symptoms of the disease. Furthermore, NMDAR antagonists also increase Gamma power amplitude, both in human and in animal models. In this study we investigate by means of computational models how NMDAR antagonists, such as ketamine, affect the dynamics of neural networks and how the generated boosting of Gamma activity affects the network response, providing an interpretation for the observed correlation between Gamma Power and psychotic episodes.
Several preparations with sub-anesthetics doses of NMDAR antagonists have reported to produce neural excitation. Since NMDAR mediate excitatory synaptic transmission, this behavior is intriguing. Several hypothesis have been proposed to explain this apparent paradox. One of the possible explanations is that NMDAR antagonists in sub-anesthetics doses act preferentially on inhibitory neurons, increasing network activity indirectly by means of desinhibition. Even though some contrasting results have been observed, this interpretation has been supported experimentally by several works. Network excitability have also been reported to increase in schizophrenic patients, and its increase in sensory and association cortex have been correlated with hallucinations. Another important effect of NMDAR antagonists in subanesthetics doses is the increase of Gamma-band activity. These observations were reported in human, monkeyand rats, both during cognitive tasks or free movement. The network model developed in the present work (see Methods) is able to reproduce both of these features (increase of network excitability and increase of Gamma power). Figuredepicts the network behavior with respect to the to different NMDA synaptic strengths, Q N M DA , in excitatory Regular Spiking (RS) and in inhibitory Fast Spiking (FS) cells. We mimic the block of NMDA channels due to the action of NMDAR antagonists by decreasing Q N M DA in RS and FS cells according to Figure(see Methods). Points of higher synaptic strengths are associated with healthy conditions, while points with lower synaptic strengths are associated to pathological conditions supposedly similar to the schizophrenic brain. The network dynamics for two sets of NMDA synaptic strengths are shown in Figureand Figureby means of a Raster Plot. As the synaptic strengths of NMDA channels decreased (higher concentration of NMDAR antagonists), the firing rate of excitatory RS cells increased while the firing rate of inhibitory FS cells decreased (Figure). In addition, the Gamma power of the population activity (see Methods) presented an increase (Figureand).
We investigated how the decrease of NMDA synaptic strength changed the network dynamics and its capacity to respond to external stimulus. While network excitability is related to an overall increase of spiking activity, network responsiveness relates to the network capacity to react to a certain stimulus, producing additional spikes then the ones generated by spontaneous activity. These two dynamical measurements (excitability and responsiveness) are not always congruent, meaning that it is possible to observe an increase in excitability but a concomitant decrease in responsiveness. Network responsiveness was defined as the difference between the total number of spikes generated by the whole network in the presence and in the absence of the stimulus (see Eq 6). We measured network responsiveness at different levels of NMDAR block for different stimulus amplitudes (Figure). The stimulus consisted of a variation in time of the external Poissonian drive, in a Gaussian manner (see Methods). Network responsiveness in RS cells increased with the increased level of NMDAR block, while the responsiveness of FS neurons decreased. In this case, both, network excitability and network responsiveness, behave in the same direction. The increase of network responsiveness can be understood from Figure. The NMDA receptors block depolarizes RS cells, while FS neurons are overall hyperpolarized. For weak levels of NMDA receptors block, no or weak depolarization is observed in FS cells, while for strong levels of NMDA block a significant hyperpolarization is observed.
Gamma oscillations are believed to be involved in information processing, and have been associated to different high-level cognitive functions, such as memory, perception, attention, focused arousaland prediction. In parallel, studies with schizophrenic patients have reported a positive correlation between psychotic symptoms and the power of Gamma oscillations. In contrast, Asynchronous-and-Irregular (AI) statesare usually associated to conscious states, being observed during awake and aroused states. This regime are characterized by irregular and sustained firing with very weak correlations. In a previous studywe reported that AI states, in comparison to oscillatory states in Gamma band, provide the highest responsiveness to external stimuli, indicating that Gamma oscillations tend to overall diminish responsiveness. This observation could indicate that Gamma rhythms present a masking effect, conveying information in its cycles on spike timing at the expense of decreasing the strength of the network response. In the present study, we compare AI and Gamma states at different levels of NMDAR block. Figuredepicts the responsiveness of RS neurons, with respect to different stimulus amplitudes (same protocol as Figure), for different ensembles of NMDA synaptic strengths. In agreement with Figure, parameter sets in which NMDA synaptic strengths are decreased (mimicking the action NMDAR antagonists) correspond to regions of the parameter space with higher responsiveness. For example, Q N M DA F S = 0.4 nS and Q N M DA F S =0.36 nS displayed higher responsiveness then the networks in which the NMDA synaptic strengths wrere Interestingly, in both conditions, responsiveness in AI states were always superior to the one in Gamma. This result was also observed in a similar model in the obscene of NMDA channels. This example illustrates a general tendency, which was also observed with other parameter sets.
In this work, we used computational models to investigate the effect of psychotic drugs such as ketamine in cerebral cortex, and how gamma oscillations relate to these effects. Our findings are (1) NMDA receptors antagonists modulate the rhythms produced by a simple network model consisting of two distinct cell types, RS and FS cells, which generate Gamma oscillations by means of the PING mechanism. This modulation is obtained assuming that the NMDAR block predominantly affects interneurons. (2) The boosted gamma oscillations following partial block of NMDA receptors, was accompanied by an increased responsiveness to external inputs.This increase of responsiveness could also be seen for asynchronous states, with no apparent gamma. We discuss below the implications of these findings. A first prediction of the model is that it was necessary that the antagonism affects predominantly NMDAR receptors on interneurons. This feature is supported by a number of observations. Intuitively, if the NMDAR block would occur predominantly on excitatory cells, then it is difficult to see how diminishing excitation could augment the activity and excitability of the network. This long-standing question was resolved recently by finding that indeed, NMDAR antagonists primarily affects NMDA receptors on interneurons. It was observed that the application of Ketamine or MK-801 in subanesthetic doses leads to an increased activity of glutamatergic neurons both in cortexand in hippocampus, and that this increase of glutamatergic activity is a consequence of the disinhibition of GABAergic neurons. In addition, it has also been reported in hippocampus that inhibitory neurons are more sensitive to NMDAR antagonists than glutamatergic neurons. Thus, our model completely supports these findings, and could reproduce the increase of Gamma power induced by NMDA receptor antagonists. On the other hand, contrasting results also exist. For example,argue that NMDAR have less impact on the activity of inhibitory neurons than on the one of excitatory neurons, since they and other authors observed that NMDAR block depressed large EPSP-spike coupling more strongly in excitatory than in inhibitory neurons. The second finding, which is probably the main finding of our study, is that the network has a marked increased responsiveness under the boosted Gamma condition. This increased responsiveness could be tested experimentally either in vitro, by testing the response of cortical slices with and without application of NMDAR antagonists, or in vivo, by monitoring their response following administration of NMDA antagonists. The third finding is that the increase of responsiveness is not specific to gamma oscillations, because it was also present for asynchronous states with no apparent gamma. The underlying mechanism is that the antagonism of NMDA receptors produce an overall depolarization of RS cells, and hyperpolarization of FS cells. Consequently, there is an increase of responsiveness of RS cells, with a corresponding decrease for FS cells, as we observed. In this model, the increase of responsiveness is due to the depolarizing effect on RS cells, and are not due to gamma oscillations. Indeed, the highest responsiveness was seen for asynchronous states, also in agreement with a previous modeling study.
Our model exhibits several interesting properties that can be related to pathologies. First, the model provides a possible explanation for the symptoms associated to ketamine and others NMDA receptor antagonists, such as hallucinations. The enhanced responsiveness produced by antagonizing NMDA receptors may explain exacerbated responses to sensory stimuli, which may be related to phenomena such as altered perception or hallucinations. Indeed, it is well documented that ketamine produces hallucinations together with a marked increase of gamma oscillations. Besides hallucinations, the model seems also a priori con- sistent with the previously reported role for FS neurons in schizophrenia. Post-mortem analysis of schizophrenic patient brains have shown a reduced expression of parvalbumin (PV) and GAD67. In parallel, genetic ablation of NMDA receptors in PV-positive interneurons in rodents mimics important behavioraland phenotypical features of of the disease (reduction of GAD67, increase of neuronal excitabilityand increase of spontaneous Gamma power). These observations support the idea that the hypofunction of NMDA receptors in PV-positive interneurons are specially important in this illness. However, NMDA receptors are expressed in both GABAergic and glutamatergic neurons, and it still remains unclear in which types of cells the NMDA receptor hypofunction causes schizophrenia. Some works reported conflicting results and have questioned the hypothesis that PV-positive Fast Spiking neurons play a role in Schizophrenia. In our model, the effect of NMDAR antagonists is to increase excitability due to desinhibition, consistent with a number of experimental observations. This increased excitability is accompanied by a Gamma power increase, as also found in experiments with ketamineor in schizophrenic patients. The model could reproduce all these experimental observations only assuming a larger decrease of the NMDA synaptic strengths in FS cells than in RS cells (see Figure). These results support the idea sustained by some authors, that PV-positive Fast Spiking inhibitory neurons play a key role in schizophrenia. Another modeling study also stressed the importance of NMDA channels into FS neurons. Thus, models support the view that the hypofunction of NMDA receptors on FS cells could explain a number of features typical of schizophrenia, such as anomalous responses and boosted gamma oscillations.
Neural units are described by the Adaptive Exponential Integrate-And-Fire Model (Adex). In this model, each neuron i is described by its membrane potential V i , which evolves according to the following equations: (1) where C is the membrane capacitance, g L is the leakage conductance, E L is the leaky membrane potential, V th is the effective threshold, ∆ is the threshold slope factor and I Syn i (t) is postsynaptic current received by the neuron i (see next section). The adaptation current, described by the variable w i , increases by an amount b every time the neuron i emits a spike at times t j and decays exponentially with time scale τ w . The subthreshold adaptation is governed by the parameter a. During the simulations, the equation characterizing the membrane potential V i is numerically integrated until a spike is generated. Formally this happens when V i grows rapidly toward infinity. In practice, the spiking time is defined as the moment in which V i reaches a certain threshold (V th ). When V i = V th the membrane potential is reset to V rest , which is kept constant until the end of the refractory period T ref . After the refractory period the equations start being integrated again. In the developed network two types of cells were used: Regular Spiking (RS) excitatory cells and Fast Spiking (FS) inhibitory cells. The cell specific parameters are indicated in Table.
The post-synaptic current received by each neuron i is composed by three components: two excitatory, referent to AM P A and N M DA synaptic channels, and one inhibitory, referent to GABA A channels. in which E AM P A = 0 mV, E GABA A = -80 mV and E N M DA = 0 mV are the reversal potentials of AM P A, GABA A and N M DA channels. While the AM P A and GABA A -mediated currents are fast, the NMDA-mediated are considerably slower and present a complex relation with respect to the membrane potential. This complex relation , due to magnesium block, is accurately modeled by the phenomenological expression B(V): where [M g 2+ ] o = 1 mM is the external magnesium concentration (1 to 2 mM in physiological conditions). Because of the fast dynamicas of AM P A and GABA A channels, their synaptic conductances (G X with X=AM P A , GABA A ) are usually modeled to increase discontinuously by a discrete amount Q X , every time a presynaptic neuron spikes at time t k , and to subsequently decay exponentially with a decay time constant τ X decay according to the following equation: In which, k runs over all the presynaptic spike times. The synaptic time constantes used for AM P A and GABA A synapses are τ AM P A decay = 1.5 ms and τ GABA A decay = 7.5 ms. NMDA channels synaptic conductances, G N M DA , because of their slow dynamics, are usually modeled as a biexponential function characterized by a rise time constant, τ N M DA rise = 2 ms, and a decay time constant τ N M DA decay = 200 ms, according to the following equation: In which, Q N M DA i is the synaptic strength of the NMDA synapse towards the neuron i, α= 0.5/ms and x(t) is an auxiliary variable. The k runs over all the presynaptic spike times. Both, s(t) N M DA and x(t), are adimensional. Synaptic strenghs of N M DA synapses (towards RS and FS neurons) were chosen according to the parameter search expressed in Figure, while the synaptic parameters of AM P A and GABA A synapses were chosen according to previous works(Q AM P A = 5 nS and Q GABA A = 3.34 nS). All synapses (AM P A, GABA A and N M DA) were delayed by time of 1.5 ms. With these choice of parameters the NMDA/AMPA charge ratio in the network is on average higher in RS cells then in FS cells (see Figure), in agreement with experimental measurements in prefrontal cortex of adult miceand rat.
The network developed in this work is composed of 5000 neurons (4000 RS and 1000 FS). Each neuron (RS or FS) was connected randomly to every other neurons in the network with a probability of 10%, receiving on average 500 excitatory synapses (mediated by both AM P A and N M DA channels) and 100 inhibitory synapses (mediated by GABA A channels).
In addition to recurrent connections, each neuron received an external drive to keep the network active. This external drive consisted of N ext = 5000 independent and identically distributed excitatory Poissonian spike trains with a spiking frequency µ ext . These spike trains were sent to the network with a 10% probability of connection and were computed inside of the synaptic current term I AM P A , with a synaptic strength of Q AM P A Ext = 0.8 nS. For Gamma activity, the network was stimulated with a drive with µ ext = 3 Hz. For Asynchronous and Irregular activity, the network was stimulated with a drive with µ ext = 2 Hz. The external drive mimicked cortical input, like if the network was embedded in a much bigger one. To test network responsiveness, an additional external input was included in the simulations. This external input, similar to the external drive, also consisted of N ext = 5000 independent and identically distributed excitatory Poissonian spike trains, connected to the network with a 10% probability. The difference of this input was its firing rate time dependence (µ ext (t)). The spiking frequency of the spike trains varied in a Gaussian manner, with a standard deviation of 50 ms and variable amplitude. These spike trains were computed inside of both synaptic current terms I AM P A and I N M DA , with a synaptic strengths of Q AM P A Ext = 0.8 nS, and Q N M DA Ext RS and Q N M DA Ext F S as indicated in each case.
In this work, we mimic the effect of NMDAR antagonists by changing the value of the NMDA synaptic weights Q N M DA . In Figurea possible trajectory in the parameter space generated by the action of NMDAR antagonists is depicted. This is the same trajectory indicated in Figure.
All neural networks were constructed using Brian2 simulator. All equations were numerically integrated using Euler Methods and dt=0.1 ms as integration time step. The codes for each one of the three developed networks are available at ModelDB platform.
To measure the global behavior of the neuronal population, we used a simulated Local Field Potential (LFP). This LFP was generated by the network, by means of a recent method developed by. This approach calculates the LFP by convolving the spike trains of the network with a Kernel that have been previously estimated from unitary LFPs (the LFP generated by a single axon, uLFP) measured experimentally. Since this method assumes a spatial neuronal displacement, to be able to apply it to our simulations, we randomly displaced part of the network (50 neurons) in 2-D grid, assuming that the electrode was displaced on its center and was measuring the LFP in the same layer as neuronal soma. The program code of the kernel method is available in ModelDB (), using python 3 or the hoc language of NEURON.
The Power spectrum of the simulated LFP was calculated by means of the Welch's method, using a Hamming window of length 0.25 seconds and 125 overlapping points. We used the Python-based ecosystem Scipy function signal.welch to do our calculations.
The synaptic charge (AMPA or NMDA) of each neuron is defined as the area under the curve of the synaptic current (shaded areas of Figure A or B), which was calculated from the presynaptic input time until 10 ms after it.
The level of responsiveness (R) of a due to an stimulus (S) in a time window of duration T , is defined as the difference between the number of spikes generated by the whole network due to an (N S spikes ) and the total number of spikes generated in the absence of the stimulus (N spikes ), normalized by the network size (total number of neurons N n ) and the duration of the time window T .
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