Dynamics of spontaneous activity in random networks with multiple neuron subtypes and synaptic noise

Our work is based on a recent model that describes self-sustained oscillations across high (up) and low (down) global activity states (Tomov et al. 2014, 2016). This is the standard random network model where directed connections between every two nodes exist with a fixed probability p. To keep cortical sparseness we have chosen a low connection probability p = 0.01 and size N = 2¹⁰. This renders the expected number of incoming connections per node (average in-degree) p(N − 1) ≈ 10. The network is mixed: it includes both excitatory and inhibitory nodes. The sizes of excitatory and inhibitory subpopulations are taken in the proportion 4 : 1 (Brunel 2000). Each network node is a neuron modeled by the Izhikevich formalism (Izhikevich 2003) with parameters that ensure diverse dynamics on the individual level. Every neuron is described by two variables: voltage v(t) and membrane recovery variable u(t), which follow the coupled differential equations with a fire and reset rule. Every time when v(t) assumes the threshold value v(t) = v_peak, both variables are instantaneously updated: Our choice of the Izhikevich neuronal model is based on its ability to mimic, by means of setting the appropriate values of parameters a, b, c, d, the behavior of neurons from different electrophysiological classes (Nowak et al. 2003; Contreras 2004). Among those, we concentrate in this study on the excitatory regular spiking (RS) and chattering (CH) neurons, and on the inhibitory fast spiking (FS) and low-threshold spiking (LTS) neurons. Figure 1 shows examples of individual dynamics for different classes: the neuronal types differ in frequency, adaptation, and in rheobase current. We consider network compositions where all inhibitory neurons belong to the same class: all of them are either of the LTS type or of the FS type. In the excitatory subpopulation we take the case when all neurons belong to the RS type, and the case when the RS neurons are mixed with CH. A thorough discussion of different aspects of the Izhikevich neuron model can be found in Izhikevich (2007).

The last term in the first equation of Eq. (1) describes the synaptic current which, for a neuron j, reads: The current is controlled by conductances \(G_{j}^{\text {ex/in}}\) and reversal potentials E_ex/in, responsible for excitatory/inhibitory effects. Whenever an excitatory (inhibitory) neuron spikes, an increment g_ex (g_in) is added to the conductances G^ex (Gⁱⁿ) of all its postsynaptic neurons; thereafter the conductances decay exponentially with time constant τ_ex/in. This is well known as a conductance based synaptic model, described by the differential equation where summation is performed over all time instants t_i of preceding presynaptic spikes. We adopt the same parameters as in Tomov et al. (2016): E_ex = 0 mV, E_in = −80 mV, τ_ex = 5 ms and τ_in = 6 ms.

The last term in Eq. (4) is the synaptic noise source. Since, for simplification, the noise sources are treated as being independent or weakly correlated, a superposition of a large number of such inputs is approximated by a simple Gaussian white noise process. We assume that ξ_j is Gaussian with zero mean and unit variance: \(\left \langle \xi (t)\right \rangle = 0\) and \(\left \langle \xi (t)\xi (s)\right \rangle =\delta (t-s)\). Note that in spite of the zero mean of the Gaussian process, the mean value of the synaptic input current \(I_{syn,j}\) stays non-zero which, in its turn, is determined by both \(G_{j}^{\text {ex/in}}\) and \(E_{\text {ex/in}}\). So, the Gaussian process only has the effect of causing displacements in the synaptic current but does not act as a driving force. Concerning the variance, since the sum of independent random normally distributed variables is normally distributed as well, the overall variance of the stochastic process for a neuron j is chosen to be proportional to the total number of excitatory/inhibitory inputs \(n_{j}\) that this neuron receives. Thereby, for neurons with different numbers of presynaptic partners, the intensity of the noisy input is different. Altogether, evolution of conductances for each neuron consists of the stochastic Ornstein-Uhlenbeck process (Uhlenbeck and Ornstein 1930) in the time intervals between the presynaptic spikes and of discontinuous jumps upwards of the size \(g_{\text {ex/in}}\) at the instants of arrival of those spikes. This stochastic model, similar to the point-conductance model described in Destexhe et al. (2001), has its power-spectral density and variance completely determined (Gillespie 1996). In distinction to Destexhe et al. (2001), in our case randomness is generated within the synapses and is, in general, non-Poissonian.

The complete set of parameter values used in the simulations of this study is summarized in Table 1. Note that that the values of parameters \(\alpha , \beta , \gamma \) in the voltage equation are shared by all neuronal types.

In this subsection, we introduce neuron and network measures that will be used below for characterization of the results.

We start by defining the network time-dependent firing rate as where the spike train \(x_{j}\) for each neuron j is viewed as a series of \(\delta \) functions: \(x_{j}(t) = {\sum }_{{t^{f}_{j}}} \delta (t-{t^{f}_{j}})\), with \(\{{t^{f}_{j}}\}\) being the set of times when neuron j fired. We fix the time window Δt = 1 ms.

We will use two power spectra: the spike train power spectrum and the voltage time series power spectrum. The first one is defined for each neuron j as where \(\tilde {x}_{j}(f)\) is the Fourier transform \(\tilde {x}_{j}(f) = {{\int }_{0}^{T}}dt\,e^{2\pi ift}x_{j}(t)\), \(\tilde {x}_{j}^{*}\) is the complex conjugate and T is the length of the time interval. Note that 〈.〉 represents an ensemble average. The power spectrum of the voltage time series is obtained in the same way, replacing in Eq. (6) the spike train \(x_{j}(t)\) by the voltage time series \(v_{j}(t)\). In the case of the spike train power spectrum, the units are 1/s whereas the units of the voltage spectrum are mV²/Hz.

An average over a subset that includes K neurons renders the average power spectrum:

We quantify the degree of oscillatory activity in the network via the spectral entropy \(H_{s}\) (Blanco et al. 2013; Sahasranamam et al. 2016). Spectral entropy is computed from the time-dependent firing rate Eq. (5) as where \(N_{b}\) is the number of frequency bins and \(S_{rr}(f_{k})\) is the value of the normalized (i.e. \({\sum }_{k} S_{rr}(f_{k}) = 1\)) power spectrum of the network time-dependent firing rate \(r(t; {\Delta } t)\) at the k th bin. In our simulations we use \(N_{b}\)= 1000. In the case of broadband noise activity, the power spectrum of the network firing rate is flat and the spectral entropy is maximal: \(H_{s}= 1\). If, in contrast, all power is concentrated at one frequency, a case of single-frequency network oscillations, the spectral entropy vanishes: \(H_{s}= 0\).

To quantify the degree of synchrony in the network, we use the phase locking value (PLV ) which is a standard measure to evaluate phase synchronization (Lachaux et al. 1999; Celka 2007; Rosenblum et al. 2001; Aydore et al. 2013; Lowet et al. 2016). Unless otherwise stated, the time average used to calculate the PLV is always taken over a simulation interval of T = 2000 ms. We define the PLV as the average over K neuron pairs and T sample time points: where \({\Delta } {\Phi }_{xy}(t)\) are the phase differences \(\rho _{x}{\Phi }_{x}(t)-\rho _{y}{\Phi }_{y}(t)\) from two randomly chosen spike-trains \(\left (x(t),y(t)\right )\) that are obtained using the Hilbert transform. The values \(\rho _{x}\) and \(\rho _{y}\) define the frequency ratio and, expecting similar firing rates, we set \(\rho _{x}=\rho _{y}= 1\). The PLV is bounded between 0 (asynchrony) and 1 (synchrony).

In our simulations, we constructed parameter space plots of the synchrony index PLV (like the ones shown in Results) for different numbers K of neuron pairs and observed a saturation in the plots for increasing values of K above 50. This indicates that PLV becomes independent of the number of neuron pairs for K ≥ 50. To ensure this independence, in computations we took K = 60.

Numerical integration of the differential equations was performed by means of the Heun algorithm (Mannella 2002). We used C++ to write the computational code, and Matlab and xmgrace to visualize and analyze the results.

To single out the effects caused by the introduction of synaptic noise, we first characterize the system in the non-perturbed state, i.e. in the absence of noise. Below, we refer to this case as the deterministic setup.

At the chosen parameter values the global state of rest is stable. Since in the deterministic setup no activity can be excited from that state without an initial disturbance, we start simulations by applying brief electric stimulation to arbitrarily selected neurons. Different stimuli are constructed by varying

The initial kick provided by brief stimulation has a sole role to put the system into a state other than rest. After the stimulation ends, the network is left to evolve freely and its dynamics is recorded. Eventually all trials end up in the state of rest. In most cases evolution is not a straightforward decay but a long dynamical transient; its duration strongly (by several orders of magnitude) varies, depending on the initial conditions. On discarding the cases where the free activity was shorter than 400 ms, we are left with a set of trials in which the network displayed long-living self-sustained activity; duration of the latter stage justifies a closer look at its intrinsic characteristics.

We have studied different combinations of the conductance increments (g_in, g_ex) and observed rather distinct behavior as shown in Fig. 2. The choice of \(g_{\text {ex}}\) and \(g_{\text {in}}\) directly affects the network balance and shapes thereby its dynamics (Vogels et al. 2005b).

Depending on the ratio \(g_{\text {in}}/g_{\text {ex}}\), the self-sustained activity displayed by the network belongs to one of two categories outlined in Tomov et al. (2014). The first one, shown in the left column of Fig. 2, is a relatively constant network activity state where neurons spike in an asynchronous and non-oscillatory fashion. For the given example, this is confirmed by the high value of the spectral entropy (H_s = 0.87) and the low phase locking value (PLV = 0.39). The reason for the constant network activity can be seen from the behavior of the voltage traces for two randomly selected neurons at the bottom of the left column of Fig. 2: the neurons fire irregularly, but their firing rates are so high that the collective activity is constant.

The second category, shown in the right column of Fig. 2, is an oscillatory state (H_s = 0.39) characterized by regular periods of high mean firing rate intercalated with periods of very low firing. The average voltage indicates that the bulk of neurons is fluctuating between depolarization and hyperpolarization. The PLV for this state is higher (PLV = 0.60) indicating that neurons fire/stay silent with a higher degree of synchrony. Voltage traces for two randomly picked neurons (see bottom plot in the right column of Fig. 2) show bursts of closely spaced spikes during high activity phases intercalated with periods of hyperpolarization below the reset value during low activity phases. This behavior was explained by us earlier (Tomov et al. 2016) in terms of the dynamics of the recovery variable u in the single neuron phase plane of the Izhikevich neuron.

The example given in the middle column of Fig. 2 illustrates the transition between the two above categories. This transition occurs when the inhibitory synaptic increment overcomes the excitatory synaptic increment as reported in Tomov et al. (2014). The network activity in the transition region looks as a mixture of constant and oscillatory activity, with intermediate values of the synchrony (PLV = 0.45) and oscillatory activity (H_s = 0.56) indexes. Voltage traces for two randomly chosen neurons (bottom of middle column) show high firing rates like in the first category (a tendency for constant activity), but now there are short periods of activity break like in the second category (oscillatory activity).

Naively, \(g_{\text {in}}/g_{\text {ex}}= 4\) may seem to be a balanced situation, as reported elsewhere (Brunel 2000). Here, however, we are dealing with neurons from different electrophysiological classes, and their firing rates differ as well. In addition, we are using a conductance based synaptic model where the synaptic current is voltage-dependent. In that sense, the mean time-averaged synaptic input for a given neuron j can be roughly estimated as where \(C_{\text {ex/in}}\) are the numbers of excitatory/inhibitory inputs to neuron j, \(\nu _{\text {ex/in}}\) are the mean firing rates of the excitatory/inhibitory populations, and \(\langle v \rangle \) is a representative voltage. The expression in Eq. (10) elucidates that the notion of “balance” is subtle, and its reduction to just \(g_{\text {ex/in}}\) and \(C_{\text {ex/in}}\) may be misleading. Usually, when LIF neurons are considered, equal mean firing rates of excitatory and inhibitory neurons, as well as equal relaxation times \(\tau _{\text {ex,in}}\) are assumed, hence the balance requires only that \(g_{\text {in}}/g_{\text {ex}}=C_{\text {ex}}/C_{\text {in}}\), which, in the widely studied situation with the number of excitatory connections four times higher, results in \(g_{\text {in}}/g_{\text {ex}}= 4\). In contrast, in a network like ours, with \(\nu _{\text {in}}>\nu _{\text {ex}}\), there is no balance at \(g_{\text {in}}/g_{\text {ex}}= 4\), instead there is a voltage dependent input current: if \(\langle v \rangle \) is depolarized (hyperpolarized), negative (positive) currents drive the neuron.

Altogether, these preliminary examples confirm that the deterministic setup, depending on the ratio \(g_{\text {in}}/g_{\text {ex}}\), is able to generate oscillatory or constant activity. In the following, we concentrate on the oscillatory situation, when inhibition overcomes excitation.

In Fig. 3 we present an exemplary simulation in the deterministic setup and extended statistics from the set of long-lived realizations with synaptic increments \(g_{\text {ex}} = 0.15\) and \(g_{\text {in}} = 1\) (this set contains 487 simulations, thus allowing good statistics). In this case the majority of neurons oscillates between a depolarized state and a hyperpolarized state, well visible in Fig. 3B and on the bimodal distribution in Fig. 3F, computed from the entire set of simulations with varied initial stimulation. For individual neurons these preferred subthreshold membrane potentials are known as “up” and “down” states (Wilson 2008), and in the context of the ensemble of neurons it seems natural to view these two states as collective “up” and “down”, respectively. As seen in Figs. 3A-E, a typical period of oscillations is close to 100 ms (f ≈ 10 Hz).

For a typical neuron in the ensemble, Figs. 3D-E illustrate the temporal evolution of the voltage and the membrane recovery variable, respectively, during the same simulation. There is strong correlation between firing of this neuron and the periods of high activity of the whole network, although some other neurons also fire when the network activity is low. These latter are inhibited during the high activity epochs and become disinhibited when the overall network activity is low. We have shown elsewhere the importance of this disinhibitory effect to sustain the long-lived activity of the network in the oscillatory situation (Tomov et al. 2016).

Remarkably, not only the voltage series in Fig. 3D features two different states (a hyperpolarized one and a depolarized one) but also the membrane recovery variable, which clearly grows when the network activity is high and slowly relaxes when the activity is low. This is a global phenomenon: in all simulations there are peaks of the variable u. In the distribution shown in Fig. 3 G, the maximum is broad due to the time-scale separation of the variables: u is slower than v and its relaxation takes much longer. In Tomov et al. (2016) we have shown the importance of the recovery variable for the breakdown of global high-activity epochs, which produces the up and down oscillatory pattern.

Figure 3H presents a histogram of durations in collective up and down states. The term “up” refers here to different states in which the network activity is above 20% of its average value, while the voltage for the majority of neurons is at a depolarized value. A collective “down” state is identified whenever the bulk of the neurons reaches a hyperpolarized state close to \(-80\) mV.

Recall that eventually the system ceases to oscillate, and voltages of all neurons invariably converge to the rest value.

Introduction of synaptic noise drastically changes one important aspect, both in the individual and in the collective dynamics: the state of rest, albeit formally stable, ceases to be the ultimate attractor. A neuron is an excitable system, and in the noisy setup it is just a matter of time when a sufficiently strong fluctuation (or a cumulative effect of many fluctuations) drives it across the spiking threshold. For an ensemble this implies disordered sporadic firing of its members, which, under favorable conditions, can turn into ordered collective activity. If deterministic aspects dominate in dynamics, this activity will temporarily end in the state of rest, only to be recreated by new fluctuations.

Here we describe various collective states induced in the network by synaptic noise. Experience gained from the study of the deterministic setup allows us to expect that, along with the synaptic noise intensity D, the crucial parameter in this context is the ratio \(g_{\text {in}}/g_{\text {ex}}\): the proportion between inhibitory and excitatory synaptic strengths (Brunel 2000; Girones and Destexhe 2016). We start by exploring the behavior of the spectral entropy \(H_{s}\) and the synchrony measure PLV in the two-dimensional diagram spanned by parameters \(g_{\text {in}}/g_{\text {ex}}\) and D (Fig. 6).

As seen in the diagrams in Fig. 6, both \(g_{\text {in}}/g_{\text {ex}}\) and D are responsible for shaping the activity pattern of the network. Let us begin with the diagram for spectral entropy in Fig. 6 A. For weak synaptic noise (\(D \lessapprox 5 \times 10^{-6}\)) the network displays non-oscillatory behavior independently of the ratio \(g_{\text {in}}/g_{\text {ex}}\). For the narrow horizontal band defined by \(5 \times 10^{-6} \lessapprox D \lessapprox 10^{-5}\), the state of the network is oscillatory and the degree of oscillatory activity is higher for \(g_{\text {in}}/g_{\text {ex}} \lessapprox 2\). On the other hand, for \(D \gtrapprox 10^{-5}\) the situation is inverted and the region determined by \(g_{\text {in}}/g_{\text {ex}} \lessapprox 2\) displays non-oscillatory activity while most of the remainder of the diagram features oscillatory activity. Within this latter part of the diagram, increase of both noise and inhibitory synaptic strength lowers the degree of oscillatory activity until in the upper right corner the activity turns non-oscillatory.

Now let us turn to the diagram for the synchrony PLV in Fig. 6B. The region of weak synaptic noise (\(D \lessapprox 5 \times 10^{-6}\)) displays asynchronous behavior independently of \(g_{\text {in}}/g_{\text {ex}}\). Under such weak noise firing remains an individual event for noise-perturbed neurons, rather than a collective effect. Along the narrow horizontal band of the diagram determined by \(5 \times 10^{-6} \lessapprox D \lessapprox 10^{-5}\), the synchrony index has mostly intermediate values with a narrow high-synchrony region around \(g_{\text {in}}/g_{\text {ex}} \approx 4\). In the remainder of the diagram the behavior along horizontal scans in the diagram is roughly the same: in the entire region determined by \(g_{\text {in}}/g_{\text {ex}} \lessapprox 2.5\) the activity is asynchronous, whereas outside that region the degree of synchrony has intermediate values.

The combined information in the two diagrams of Fig. 6 is qualitatively summarized in a schematic diagram drawn in Fig. 7. The states in this diagram are denoted in accordance with two measures of network activity in Fig. 6: \(H_{s}\) quantifies the degree of oscillatory activity and PLV quantifies the degree of synchrony. Selected samples from the different regions are also displayed on the right of Fig. 7 to show the time-dependent network firing rates for the corresponding combinations of \(H_{s}\) and PLV.

The region of weak synaptic noise lies at the bottom of the diagram in Fig. 7. The type of network activity there is asynchronous non-oscillatory, already described in Section 3.2.2. It is similar to the asynchronous irregular (AI) activity observed in networks of LIF neurons (Brunel 2000; Vogels et al. 2005b). The region stretches along the full length of the horizontal axis, indicating that the generic features of the network activity for weak synaptic noise are insensitive to the ratio between excitation and inhibition.

For stronger synaptic noise the structure of the diagram in Fig. 7 is more complex. The network displays synchronous oscillatory activity within an irregular shaped region in the center of the diagram, adjoined by a narrow horizontal strip in the bottom part. This is similar to the synchronous regular (SR) type of activity found in networks of LIF neurons (Brunel 2000; Vogels et al. 2005b). In the remainder of the third of the diagram where \(g_{\text {in}}/g_{\text {ex}} < 2\) the activity is asynchronous non-oscillatory. Its pattern is similar to the constant pattern shown in the left column of Fig. 2. On the other hand, through the upper two-thirds of the diagram for \(g_{\text {in}}/g_{\text {ex}} > 2\) the activity is synchronous non-oscillatory. Thus, for very strong synaptic noise the network activity is non-oscillatory and can be synchronous or asynchronous depending on the \(g_{\text {in}}/g_{\text {ex}}\) ratio.

Finally, the diagram in Fig. 7 includes the region marked as “transition”. It contains most of the right third of the diagram, with the exception of the regions of weak and strong synaptic noise mentioned above, and extends to the central part of the diagram where it separates the synchronous oscillatory from the asynchronous non-oscillatory regions. This corresponds to a region with intermediate degrees of oscillatory activity (the greenish region in the diagram for \(H_{s}\) in Fig. 6A) and synchrony (red-orange to yellow-orange colors in the diagram for PLV in Fig. 6B). Therefore, states in the transition region should occupy intermediate position between constant and oscillatory states like the state in the middle column of Fig. 2.

Interested in the behavior of the network in the transition region, we focus here on a part of the diagram in Fig. 7 determined by \((g_{\text {in}},g_{\text {ex}})=(1,0.15)\), which implies \(g_{\text {in}}/g_{\text {ex}}\approx 6.66\), and \(10^{-5} \lessapprox D \lessapprox 10^{-4}\). This corresponds to the greenish (light orange) region on the lower right-hand side of the diagram for \(H_{s}\) (PLV ) in Fig. 6A (B). Spectral entropy and PLV here are both close to 0.5 meaning that states with intermediate levels of oscillatory activity and synchrony may be encountered.

In Fig. 8 we illustrate dynamics for the point given by \(D = 1\times 10^{-5}\) and \((g_{\text {in}},g_{\text {ex}})=(1,0.15)\) in the diagram in Fig. 7. This point is in the transition region on the lower right-hand side of the diagram described above, which is characterized by intermediate values of \(H_{s}\) and PLV.

Remarkably, a typical record of a long simulation trial in this region of the diagram consists of alternating states (Fig. 8): an oscillatory one, akin to oscillations presented in the deterministic setup in Fig. 3, and a state with very low firing rates similar to the one in Fig. 5. From time to time transitions between these states occur, seemingly without any precursors. Compared to deterministic simulations, an additional feature is distinct in the histogram of mean voltage: a pronounced maximum at the state of rest. Accordingly, the temporal evolution of voltage is organized around three characteristic values, instead of two known from the deterministic setup. Three red dashed lines in Fig. 8B-C mark three relevant states; from top to bottom, they denote depolarization, the state of rest and hyperpolarization. Note that the histogram in B can be viewed as a combination of the voltage histograms from Figs. 3 and 5.

The average spectral entropy calculated over the quiescent/oscillatory states in Fig. 8 is \(H_{s} = 0.74\)/H_s = 0.37, indicating non-oscillatory activity in the first case and oscillatory activity in the second one.

We classify the observed states based on two attributes: network activity and average voltage. Like previously, the average voltage series was used to detect the up and down states (see Fig. 3H). The states close to rest are identified through very low network activity,

In terms of activity, we introduce the following distinction:

These definitions, in combination with the values of the average voltage, facilitate identification of different collective states. Certain states that look very similar on the raster plot turn out to differ in typical voltage values. For instance, both the down state and the quiescent period feature in the raster plot almost no activity, but can be easily discerned in terms of the average voltage.

In Fig. 9 we show various regimes at different values of D. Three samples corresponding to the time interval of 2 s are, from top to bottom: \(D = 0.5\times 10^{-5}\), \(D = 1.5\times 10^{-5}\), and \(D = 4.5\times 10^{-5}\), respectively. In panels A1, B1, and C1 green dots denote states with instantaneous voltage values close to the resting state, blue dots denote hyperpolarized voltage (down state), and red dots denote depolarized voltage (up state). The plot highlights the crucial role of synaptic noise level in changes of typical duration at each of these states. It is easier to generate oscillatory states (alternating between up and down states) when the network is subjected to stronger synaptic noise. In contrast, the “green” states close to rest (quiescent periods), prevalent at low synaptic noise amplitudes, occupy a much smaller proportion of time when synaptic noise becomes sufficiently intensive.

Comparison of raster plots in Fig. 9 indicates that when noise intensity D is increased, the waves of activity start to merge. This hinders identification of states, based on the raster plot alone. The spectral entropy and the synchrony index increase with the noise intensity. We expect that at very high levels of noise the activity becomes constant (synchronous non-oscillatory), with rather high firing frequencies (see the schematic diagram in Fig. 7).

In the frequency domain, variation of the noise level leads to redistribution of power in the Fourier spectra of both the spike trains and the voltage series. Figure 10 presents spectra for the same noise intensities as in Fig. 9: from top to bottom, D = 0.5 × 10^− 5, \(D = 1.5\times 10^{-5}\), and \(D = 4.5\times 10^{-5}\). All spectra were averaged over ensembles of 200 neurons, see Eq. (7). The shapes of spectral curves for spike trains and for voltage values are similar; the only noticeable difference is the somewhat faster decay at high frequencies in the voltage spectra. The left column shows mixtures of RS and LTS neurons; the right column corresponds to networks with RS and FS neurons. Under low levels of noise, spectral power is concentrated at very low frequencies, waves of collective activity are quite rare and, when they occur, they are mostly isolated events. On increasing the intensity D, waves of collective activity become more frequent whereas the periods of quiescence get shorter. During the periods of oscillatory activity, neurons are either firing at high frequency in the up state or rarely firing in the down state. This results in the increase of spectral power at low frequencies, with a distinct maximum near 10 Hz.

Comparison of left and right columns in Fig. 10 shows that spectral curves for networks with inhibitory LTS and FS neurons are similar. Comparing the peak values indicated in the panels. we see that spectral power in the networks with FS neurons is slightly higher.

Remarkably, these power spectra, computed for single neurons, bear resemblance to experimentally obtained spectral curves. In the case of the voltage spectra (Fig. 10B and D), the \(1/f^{n}\) behavior is reported in experiments on up-down states with n in the range 1 to 3 (Bédard and Destexhe 2009; Millman et al. 2010; Baranauskas et al. 2011). Furthermore, our results match the experimental observation that the spike-train power spectra have striking differences in comparison to the voltage-series power spectra (Bair et al. 1994).

In the spike-train power spectra (Fig. 10A and C), there is no \(1/f^{n}\) scaling. As the noise intensity D is raised, the value related to the zeroth frequency bin of the spectra decreases. This indicates that irregularity is becoming less apparent given that \(\lim \limits _{f\to 0}\bar {S}(f)\) is related to the Fano factor which is a measure of irregularity (Middleton et al. 2003; Pena et al. 2018).

Regarding the \(1/f^{n}\) scaling, observed both experimentally and theoretically (Beggs and Plenz 2003; Kinouchi and Copelli 2006), in our case we see that noise acts upon the scaling (cf. n values in Fig. 10B, D). It has been shown elsewhere (Baranauskas et al. 2011) that the shape of up-down transitions in the membrane potentials could be a determining factor for modulation of the \(1/f^{n}\) scaling with \(n = 2\). Our observations provide support to this experimental evidence. At unbounded growth of D, transitions should vanish, and, as a consequence, n decreases.

Additionally, increase of noise shifts the peak values and peak frequencies in both spike-train and voltage power spectra; compare the peak values in different subpanels. The existence of spectral differences where peaks becomes apparent or not is well known to be present in the cerebral cortex during different states such as slow wave sleep and wake (Buzsaki 2006).

We have seen that synaptic noise enforces alternation of collective states and influences durations of stay in each of them. Below, we explain how the dynamics of a single neuron, embedded in the synaptic noise setup, is reflected in the collective properties of activity, how the transitions are affected by the composition of the network, and how the picture changes at different levels of noise.

A deeper understanding of the single neuron behavior in the synaptic noise setup can be gained from analysis of the course of its phase plane dynamics during the simulation. Setting the derivatives \(\dot {v}\) and \(\dot {u}\) in Eq. (1) to zero renders the nullclines of the voltage and the membrane recovery variable which we denote below as \(\bar {u}\) and \(u^{*}\), respectively. with \(\bar {u}\) being a (time-dependent) quadratic parabola and \(u^{*}\) a straight line. Synaptic noise enters this configuration implicitly, through its contribution to the current I.

Under the employed parameter values (see Table 1 above) and \(I = 0\), the nullclines intersect in two points of the phase plane. These points correspond to equlibria of the system; the left of them is stable: without input current, neuron exhibits no activity. When the instantaneous value of the current is increased, the nullcline \(\bar {u}\) is shifted upwards on the phase plane, and the equilibria move toward each other. At the value \(I_{\text {sn}}(t)=\frac {(\beta -b)^{2}}{4\alpha }-\gamma \) they merge and disappear in a saddle-node bifurcation. Absence of equilibria is sufficient to ignite a spike: the voltage grows until it reaches the threshold. In fact, if the value of the parameter b exceeds that of the parameter a (this holds for all four considered neuronal types), spiking starts at even weaker current: at \(I_{H}=\frac {(\beta -b)^{2}-(a-b)^{2}}{4\alpha }-\gamma \) the subcritical Andronov-Hopf bifurcation takes place, the equilibrium loses stability and the solutions spiral out from its vicinity toward the spiking threshold. Recall that the values of \(\alpha ,\,\beta ,\,\gamma \) are common for all neuronal types (cf. Table 1). Hence, the onset of spiking at \(I_{H}\) is dictated for each type of neuron by the pertaining a and b (the remaining parameters c and d characterize the reset and are irrelevant in this context: a neuron that has made it to the reset, is already in the spiking state).

Evolution of every individual neuron is governed by its instantaneous location on the phase plane with respect to the nullclines; its dynamics is affected not only by its own state, but by the time-dependent (due to external and synaptic currents) position of the nullcline \(\bar {u}\). This allows us to see the collective dynamics from the local point of view of its individual participant; for it, the rest of the network is a background mechanism that moves the nullcline \(\bar {u}\) upwards and downwards.

Remarkably, this motion is not always negligible in comparison to dynamics of the neuron on the phase plane: on arrival of synaptic input, the nullcline \(\bar {u}\) is swiftly shifted in the vertical direction. Sometimes this leads to spectacular effects: a rapid fall of \(\bar {u}\) may drag it across the instantaneous position of the neuron on the plane and thereby halt and reverse the developing action potential. Such events, however, are seldom in a network like ours with its moderated connectivity, therefore most of the time the vertical displacements of the nullcline \(\bar {u}\) stay noticeably slower than the motion of the neuron.

With this local view in mind, we present in Figs. 11 and 12 the same simulation as in Fig. 8 focusing on the individual dynamics of two representative neurons, arbitrarily picked among the populations of, respectively, the neurons that fire only during the active periods and the neurons that fire throughout all stages of evolution. As we will see, distinctions in the behavioral patterns can be traced down to the phase planes of the neurons.

We begin from the neuron # 240 which fires only during the active periods, showing it in the time range between 3800 ms and 4400 ms. We split this range, which contains both active and quiescent states, into 6 smaller intervals \({\Delta } t_{i}\), each one of either 50 or 100 ms duration. The upper panel in Fig. 11 shows the entire range and its breakdown into the set of \({\Delta } t_{i}\). The lower panels present for every \({\Delta } t_{i}\) the voltage series and the trajectory on the phase plane. Notably, in the hyperpolarized (down) state below reset, the neuron typically is close to the instantaneous location of \(\bar {u}\), hence its motion is slow.

We summarize our observations as follows (for a clearer visualization of the moving trajectory we refer to the video in Online Resource 1):

The sequence of events in Fig. 11 discloses a major role of the membrane recovery variable u both in the transition from up state to down state and in the subsequent initiation of the new active phase by the noisy input. Because of high tonic firing during an up state, the total synaptic current into a neuron like #240 is very intense and roughly constant (its fluctuation amplitude depends on the synaptic noise level). Hence, the nullcline \(\bar {u}\) stays close to its highest position in the u-v diagram while the neuron climbs toward it due to the increments received by its recovery variable u after each spike. Finally, the neuron gets inside the parabolic nullcline \(\bar {u}\), has its firing probability decreased and eventually stops firing. The fact that the whole network enters a down state when this happens suggests that most neurons behave like #240, i.e. they dominate dynamics in the network. Excursion of the neuron to the left from the reset line while it is inside the parabola \(\bar {u}\) is the mechanism responsible for the hyperpolarized voltages seen in the down states of oscillatory regimes both in the deterministic (cf. Fig. 2) and noisy (cf. Fig. 8) setups. During a quiescent period, the nullcline \(\bar {u}\) is dragged to the bottom of the diagram putting the neuron close to rest. This explains the absence of hyperpolarized voltages during quiescent periods (cf. Fig. 8). In this situation the neuron is also close to the nullcline \(u^{*}\), so its eventual high jump to the region of the diagram below the nullcline \(u^{*}\) makes the neuron fire again and a new active period begins.

The behavior of the neuron #240 in Fig. 11 somewhat mimics the overall behavior of the network: it is highly active during up states of active periods and silent during down states of active periods and quiescent periods. In the following, we will refer to neurons of this type as “typical” in the sense that they represent the behavior of the majority of the network nodes.

The firing pattern of typical neurons is contrasted by the behavior displayed in Fig. 12. There, we show dynamics of the neuron #69, chosen because of its atypical behavior: it fires at all stages: in the up and down states of the active period and during the quiescent period. Dynamical features of this neuron are complementary to the ones of the typical neuron in Fig. 11, and a combination of the views given by them offers a deeper understanding of the mechanisms responsible for the intermittent changes between active and quiescent states.

A summary of our observations for the “atypical” neuron reads as follows (we refer the reader to the video in Online Resource 2 for a dynamical illustration of the effect):

Excitatory neurons like the one in Fig. 12, which fire at low rates at all periods, will be called here “quiet” neurons (elsewhere, in the context of the deterministic setup, we called them “moderately active neurons” Tomov et al. 2016). Quiet neurons are fewer than typical neurons; for the network of Fig. 8, they, on the average, constitute about a quarter of the population.

The sequence of events depicted in Fig. 12 underscores the importance of inhibition and synaptic noise in shaping the network activity during both down states and quiescent periods. Strongly inhibited during up states, the quiet neurons become disinhibited by the end of those states and serve as a source for most of the spikes occurring during down states and quiescent periods. Thus, the firing pattern in the down states and quiescent periods is basically due to the recurrent excitatory synaptic connections among quiet neurons. The weak noise limit (cf. Fig. 5) discloses the nature of the intrinsic activity pattern generated by the population of quiet neurons: it is highly asynchronous and non-oscillatory; remarkably, it is also weak. This confirms, on the one hand, that the population of quiet neurons is small, and explains, on the other hand, why the network activity during down states and quiescent periods is asynchronous and irregular.

Due to the weakness of intrinsic activity of quiet neurons, the likelihood that their pool can trigger a high firing (up) state in the network is low and the synaptic noise level plays a pivotal role in controlling this likelihood. At low synaptic noise level, the weak activity of the quiet neurons can restore the up state when the network is at a down state, but this can be repeated generating a sequence of up-down oscillations only for a short transient time. An example can be seen in Fig. S1. After the transient the network enters a quiescent period: a persistent low activity regime characterized by asynchronous non-oscillatory activity. When the network is in a quiescent period, the activity of the quiet neurons is too weak to start a high firing state in the network; a certain minimal synaptic noise level is necessary to trigger this state. In the absence of this minimal synaptic noise level, the network activity remains in the quiescent regime as seen in the diagrams of Figs. 6 and 7. When the synaptic noise intensity increases above minimum level, the recurrent excitation among quiet neurons gets stronger, as well as the synaptic noise inputs to typical neurons, and the probability of the network exiting a quiescent period increases.

The above discussion highlights a fundamental difference between down states and quiescent periods. In the weak synaptic noise regime, when the network activity is dictated by quiet neurons, their weak agitation is able to restore a high firing state in the network when the latter is in a down state but not when it is in a quiescent period. This phenomenon bears some similarity to the behavior observed previously by us in deterministic networks of two-dimensional nonlinear integrate-and-fire neurons in the absence of external inputs (Tomov et al. 2014, 2016). There, the network state oscillates for a transient time between up and down states, before decaying to rest (cf. the behavior of the network in the deterministic setup in Section 3.1). The decay to rest always occurs when the state of the network in its high-dimensional deterministic phase space passes through a particular region of the phase space (a “hole”) which, when represented in the two-dimensional space of average voltage \(\langle v \rangle \) and recovery \(\langle u \rangle \) variables, overlaps with the region traversed by the network when it is in a down state (Tomov et al. 2016). The analogy between down/rest state for the deterministic network without external input and down/quiescent state for the network in the synaptic noise setup suggests a further analogy between the hole in the high-dimensional phase space of the deterministic network and a hole in the high-dimensional phase space of the stochastic network. The difference is that when the network state in the stochastic high-dimensional phase space falls into its corresponding hole it escapes to a quiescent state instead of the resting state, and it can leave this quiescent state when the synaptic noise intensity is above a minimum level.

To show that the recovery variable u has a stronger impact on the cessation of activity than the inhibitory neurons, in Fig. 13 we compare the effects of this variable and the synaptic currents \(I_{syn}\) on the same neurons as in Fig. 8. In Fig. 13 we present for selected time points both variables (u, v) for 200 neurons randomly picked from the network, and their total synaptic input \(I_{syn}\).

The first row in Fig. 13 refers to an up-down transition: For \(T = 3880\) ms, which is the middle of the up state, some neurons have high values of u (due to the constant increments the u variable receives after each spike, cf. Eq. (2)) and consequently strong negative feedback. The consequence of this negative feedback is to hyperpolarize the neurons, which can be seen in the graphs for \(T = 3900\) and 3920 ms where the voltages progressively move to the left of the graph. As to the \(I_{syn}\) histograms, they are mostly dispersed around positive values (with a reduction in the dispersion as T increases) indicating a low inhibitory activity. This confirms an earlier observation that the inhibitory neurons are not the main responsibles for the up-down transition (Tomov et al. 2016). The second row in Fig. 13 refers to the down-up transition: for \(T = 3940\) ms, most neurons are hyperpolarized and the synaptic currents are sharply concentrated around zero, confirming that very few neurons (the quiet ones) are spiking, as shown in Fig. 8. As time increases, the distribution of neurons in the (v, u) plane becomes more disperse and the voltages v move to depolarized values. This indicates that the neurons are free (without negative feedback) to spike again. Meanwhile, the distribution of synaptic currents widens-up and is dominantly excitatory (although there are some inhibitory currents). The third row in Fig. 13 refers to the quiescent state: from \(T = 4000\) to 4040 ms, the variable u moves down and v moves to hyperpolarized values. As observed in Fig. 8 for the same condition, there are very few spikes. Only after about 300 ms the voltages start to grow again and firing is re-started in a new active period.

Having demonstrated in the previous section that synaptic noise affects different phases of activity, we now proceed to a quantitative description. We compute the average duration of active and quiescent periods in sufficiently long (we take the value of \(6\times 10^{5}\) ms) trials. Mean duration is an important measure to characterize and model alternating states, e.g. in the course of transitions between brain rhythms (Lo et al. 2002; Ahn and Rubchinsky 2013).

Results of simulations confirm that the duration of stay in both active and quiescent periods is affected by the synaptic noise level (Fig. 14), but in a twofold way: the growth of noise intensity lengthens active periods and shortens the quiescent ones. This implies that synaptic noise influences transitions between the states. Remarkably, the average duration of stay in the quiescent state rapidly falls at the increase of small noise amplitude but seems to reach a certain saturation at moderate noise intensities. Apparently, the minimal time that the neurons need to organize a new collective activity is dynamically constrained by the network topology and deterministic characteristic times in the phase space: in the studied case it cannot be made lower than ≈ 80-100 ms.

Depending on the network composition, action of synaptic noise upon the average duration of active and quiescent periods can be weaker or stronger. Although the same common qualitative tendencies persist, quantitative aspects depend on the types of participating neurons as well as on proportions between them. An exemplary comparison is shown in Fig. 15. Simulations with two types of inhibitory neurons indicate that the LTS neurons, compared to the FS ones, seem to postpone the termination of the active period (top left panel): at low noise the duration of oscillatory activity is higher if LTS neurons are present. This implies that inhibitory neurons influence the transition from active to quiescent period. In contrast, the duration of the quiescent period (bottom right panel) displays no dependence on the type of inhibitory neuron: the corresponding curves in the plot nearly coincide. This indicates that the transition from quiescent to active period is regulated exclusively by excitatory neurons. Indeed, since inhibitory neurons cannot excite a network, every period of stay in the quiescent period should be interrupted by an excitatory neuron, or by a group of excitatory neurons.

Introducing diversity among excitatory neurons, we observe certain quantitative changes as well. By replacing 20% of RS neurons by CH neurons, we obtain a network built of 16% CH, 64%RS and 20%LTS. This composition is much less sensitive to the action of synaptic noise. The tendency of growth of active periods under increase of noise is practically absent (see top left panel in Fig. 15), and at all values of D the average active period is shorter than the corresponding silent one. As for the latter, however, there is a systematic shift. Compared to the case when the whole excitatory population is of the RS type, in the mixture with CH neurons the mean duration of quiescent periods decreases to lower values, below \(10^{2}\) ms. This decrease is a combination of synaptic noise- and network-related effects. A quiescent period ends whenever synaptic noise or some of the few quiet excitatory neurons which fire during the quiescent period (or both) drives across the firing threshold one of the majority of typical excitatory neurons which are at rest, provided that this neuron is able to activate its postsynaptic neighbors and initiate thereby a wave of activity. If the neighbors fail to fire, the quiescent period continues. The mean time required for the first neuron to fire is the same for the RS and the CH neuron (see Section 3.2.1). However, the RS neuron issues just one isolated action potential, whereas the CH neuron generates a series of spikes, raising with each of them the conductances of its postsynaptic neighbors and creating thereby conditions for their activation and subsequent collective spiking. In this sense, a burst of a CH neuron has higher chances to initiate common activity than a spike of a RS neuron. Therefore, in a network with CH neurons the quiescent periods end earlier. This confirms our conjecture that excitatory neurons influence the length of quiescent periods.

Histograms of duration of stay in the active period in Fig. 15C1-2 show exponential distributions but are somehow fractured (cf. the logarithmic representations in the insets). Distributions of this kind have been reported previously (Duc et al. 2015; Tomov et al. 2016). In the former case the authors related cessation of activity to passages through a specific region in the phase space of their deterministic network (the “hole” mentioned above), explaining thereby the quantization of cessation times. In our case, the behavior of the system is similar. Assuming the picture of a hole in the network phase space through which the network can escape from active to quiescent state, and a synaptic noise level high enough to allow multiple transitions from quiescent to active state, the quantization of active period durations can be explained keeping in mind that an active period is made of up-down cycles, each one with the same approximate period T. Since the escapes from active to quiescent state always occur at the end of an up-down cycle, the duration of an active state can only increase by integer multiples of T. The distributions of stay duration in the quiescent period, shown in Fig. 15. D1-2, possess exponential character as well, but without a fractured shape. This can be explained by the non-oscillatory nature of the quiescent periods.

It is important to note that the results concerning influence of variation of the synaptic noise intensity on mean duration of the different regimes can be translated to other features that in the end enhance the synaptic effect. For instance, if the noise intensity is kept constant but the network size is enlarged, the same effect is expected: the mean degree will increase and consequently the synaptic effect as well (cf. Eq. (4)). Our simulations with increased network size and connection probability have confirmed this effect; an example is presented in Fig. S2 where we show that increasing the number of neurons to 5125 (5 times the standard network in this work) at constant p creates a synchronous non-oscillatory activity type. In contrast, if p is lowered, the network goes back to intermittent dynamics, although in Fig. S2 the quiescent activity is rather rare (two periods are identified in a 60 seconds simulation) and for most of the time the network remains in the active state of up/down oscillations. Finally, a variation of the network size N compensated by simultaneous change in the connectivity p, so that the mean degree \(p N\) stays constant, keeps the synaptic effects (and, hence, the prevalent types of dynamics in the network) largely unchanged.

Let us have a look at shorter timescales: what happens inside the active periods? How does noise influence the collective up and down states? In Fig. 16 we show dependence of average durations of stay in the up and down states on the noise intensity D, in the same range of D as in Figs. 14 and 15. Whereas the average stay in the down state gets shorter under the growth of noise, lifetime in the up state almost does not change.

The down state is the only state that is sensitive to the type of inhibitory neuron. There is a clear shift upwards (see red and black curves) if the LTS neurons are replaced with FS ones. This means that transitions from the down state happen faster in the presence of LTS neurons. The sensitivity of the down state can be related to the interpretation in Tomov et al. (2016) where the cessation of self-sustained oscillatory activity was assigned to passages through a small region of instability (the hole), located in the phase space close to the down state. In our current synaptic noise setup, the more noise, the higher the disturbance in the region of instability at the down state and the shorter its lifetime.

We expect the above results on intermittent transitions between active-quiescent states and the role of noise upon these transitions to stay qualitatively valid in networks based on other two-variable integrate-and-fire-type neuron models. To support this conjecture, below we apply the same procedure as in Fig. 8 to the adaptive exponential integrate-and-fire (AdEx) model (Brette and Gerstner 2005; Gerstner et al. 2014).

The AdEx model is a two-variable neuron that differs from the Izhikevich model by the equation for voltage: instead of a polynomial dependence on v, the AdEx features the exponential one. To run the AdEx network under the same conditions as the Izhikevich one without having to re-scale either the synaptic variables or the noise amplitude, we write the AdEx equations so that the input-frequency relationship and nullclines \(\bar {u}\) and \(u^{*}\) are similar to the ones from the Izhikevich model, leading to: where \({\Delta }_{T}= 30\), \(g_{L}\)= 1, \(v_{T}=-65\), and \(E_{L}=c\). The parameters (a, b, c, d) are the same as in the Izhikevich model. Along with Eq. (12), the model includes the fire-and-reset rule given by Eq. (2). A comparison of the Izhikevich nullclines to the AdEx nullclines is performed in Fig. 17A and B where one can see that, for these chosen parameters and at the resting state (I = 0), the fixed points for the two models are very close and the shape of the nullcline \(\bar {u}\) is similar.

In Fig. 17C-F we see a qualitatively close behavior to Fig. 8A-D: raster plots with \(r(t; {\Delta } t)\) indicating that active (oscillatory) and quiescent (non-oscillatory) behaviors are switching sporadically, with voltages fluctuating among three different positions (hyperpolarized, resting, and depolarized), and recovery variable u oscillating and featuring accumulation depending on the period. A noticeable difference however occurs when the average voltage in Fig. 17D for the AdEx model is compared to the one in Fig. 8B for the Izhikevich model: for the AdEx model the voltage does not stay long enough in the hyperpolarized or depolarized states to create corresponding prominent peaks in the histogram (the peak for resting voltage is much more prominent). This difference is related to the integration of the AdEx neuron model, where the growth of voltage follows an exponential law, which is much faster than the quadratic one. This effect is reflected as well in the average voltage: the peaks and troughs are sharper than those for the Izhikevich neuron model.