2.1 Mathematical setting for CTC

In this Section we describe the mathematical model for a canonical cortical neural network with intrinsic oscillatory activity in the gamma range. This model will be used to study the effects of an oscillatory input (modeling the input from a population coding for a particular stimulus) onto the network activity. In particular, we will explore phase-locking properties between emitting and receiving populations, and its implications for Communication Through Coherence (CTC) theory, that is, how changes in some input parameters are detected at the output.

Several studies on gamma-frequency oscillations have provided the fundamental neurological requirements to generate them: either from the interplay of excitatory (E) and inhibitory (I) populations or from a single inhibitory population with self-feedback. These two processes are referred to as PING (Pyramidal Interneuron Network Gamma) and ING (Interneuron Network Gamma) mechanisms, respectively, and represent the classical models to generate and explain gamma oscillations [4, 7]. In both mechanisms, inhibition plays a strong role. In this paper we focus on the PING mechanism.

To implement the PING mechanism we consider two populations of neurons, one of excitatory (pyramidal) cells and the other one of inhibitory (interneuron) cells, interconnected synaptically. With sufficiently strong and persistent external excitatory drive to the pyramidal neurons, the E-cells activate and recruit the I-cells which, in turn, send a reciprocal inhibitory feedback onto the E-cells. The inhibition causes the E-population to be less receptive to the external drive until its effect decays, starting again the E-I cycle. This mechanism generates self-sustained macroscopic oscillatory activity in the gamma range (see Fig 1).

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Fig 1. E-I cortical network and comparison between full network and reduced system.

(A) Schematic representation of a cortical neural network consisting of excitatory (E) and inhibitory (I) cells. Excitatory (resp. inhibitory) synapses are depicted by arrows with a black-filled (resp. empty) circle pointing at the receiving population. The parameter J_ab, a, b ∈ {e, i} is the connectivity strength of the b → a synapse. Besides synaptic input, each population also receives an external tonic current , a ∈ {e, i}. Additionally, both populations receive a periodic input Ap(t) (red curve), modeling the input from an oscillating neuronal population. (B) From bottom to top: Time evolution of the external input current onto the E-cells. for 0 < t < 20, , for 20 < t < 50 and , with p(t) defined in (15) with κ = 2, T = T*/2 and T* = 24.234 ms. Raster plot of 1000 randomly selected neurons (the first 800 neurons are excitatory and the last 200 inhibitory). Time evolution of the macroscopic quantities r_e and r_i obtained from simulations of the mean-field model (1)-(2) (red and blue curves, respectively) and the averaged firing rate activity of the full spiking network (black). Notice that curves lie one on top of the other, showing perfect agreement. In the latter case, the mean firing rate has been computed by averaging the number of spikes in a time window of size δt = 8 ⋅ 10⁻². Parameters: N_e = N_i = 5000, and the rest of the parameters as in (6).

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To model the E-I network we consider a neural mass model that was recently derived for networks of spiking neurons [26, 29]. It provides an exact description (in the thermodynamic limit) of the macroscopic quantities of a spiking network of quadratic integrate-and-fire neurons, namely, the mean firing rate r and the mean membrane potential V (see Methods for more details).

The model is given by a set of differential equations for the E-population, (1) and another identical set for the I-population, (2)

Here, V_a and r_a represent the mean voltage and mean firing rate of the population a ∈ {e, i}. The parameters η_a and Δ_a are, respectively, the center and width of the static distribution of inputs to the individual spiking neurons, which is considered to be Lorentzian. The time constant τ_a is the membrane time constant of the individual neurons in population a. The variable S_ab models the synaptic current from the pre-synaptic population b to the post-synaptic population a, and J_ab is the strength of the corresponding synaptic connection. The parameter corresponds to the decay time associated to fast excitatory (AMPA) synapses, and to the time constant of the fast inhibitory (GABA_A) synapses.

The total input current to the excitatory population is given by (3) and to the inhibitory one, by (4)

Notice that the terms I_e and I_i above provide the coupling between systems (1) and (2). The terms and represent the external input drive.

In this study, we are going to consider that the input current to the E and I-cells is of the form (5) where is a tonic current and g(t) = Ap(t) is a time periodic function. The parameter A modulates the amplitude of the periodic input p(t) and is chosen so that it describes the strength of the input by providing the temporal average of the stimulus (see Methods). We are going to consider a periodic function p(t) of von Mises type (see Eq (15) in Methods) to explore the effects of varying the input coherence, that is, how concentrated is the input volley around some phases of the cycle. Notice the time constant τ_e,i multiplying the periodic input, so that A describes the actual effect onto the voltage variable (see Eq (1)).

For this paper, we consider fixed the following parameters for system (1)-(2): (6) and we will vary the other ones.

In Fig 1A we present a schematic representation of the E-I neural network described by system (1)-(2) and their excitatory and inhibitory synaptic connections. In Fig 1B we show numerical simulations of the firing rate activity of the network in response to a time dependent input using the mean field reduced model (1)-(2) and the full spiking network model. Notice the perfect agreement between both models. We emphasize here the power of this reduced description, which will allow us to perform the mathematical analysis detailed in the next Sections.

We acknowledge that our model has several parameters and we have chosen a particular set . Although exploring the PING dynamics in the entire parameter space goes beyond the scope of this work, we have performed some numerical experiments to identify the qualitative role played by the most important parameters. Thus, we have observed that the presence of oscillations is a robust phenomenon and parameters mainly shape the features of the periodic orbit. For instance, changes in the synaptic strength J_ie have an impact on the firing rate activity of the I-cells, while changes in J_ei affect the oscillation frequency. Finally, time constants τ_e,i and modulate the oscillation frequency. In the next Section, we discuss in more detail the emergence of oscillations and, in particular, the role of the strength of the tonic currents to E and I cells.

2.1.1 Characterization of gamma oscillations.

Before studying the effects of a periodic perturbation onto the E-I network, we characterize the network intrinsic oscillatory activity, that is, when A = 0 in (5) and are tonic currents.

We first identify the values of the external inputs and for which the model (1)-(2) shows oscillations. With strong enough constant drive to the E-cells, the system (1)-(2) transitions from an asynchronous state of low activity (resting) to an oscillatory regime (see for instance the transition at t = 20 in Fig 1B). From the viewpoint of dynamical systems, we say that the system undergoes a bifurcation—a qualitative change in the underlying dynamics when a parameter is varied (in our case when or are varied).

We identify two relevant bifurcations in the -plane to determine the oscillatory states of the system (see Fig 2A). The asynchronous activity states destabilize through Hopf bifurcations (blue curve), which give rise to oscillations (periodic orbits) that can be stable or unstable depending on whether the Hopf bifurcation is supercritical (solid blue curve) or subcritical (dashed blue curve). Along this curve, Generalized Hopf (GH) bifurcations (also known as Bautin bifurcations) occur at the places (red circles) where the Hopf bifurcation changes from supercritical to subcritical (or vice versa) [34]. Moreover, oscillations also disappear (resp. appear) through saddle node (or fold) bifurcations of periodic orbits (purple curve). At these bifurcations, a stable and an unstable periodic orbit collide and annihilate each other (resp. sudden create). Thus, oscillations occur in the blue and orange regions enclosed by the Hopf and saddle node bifurcations. Remarkably, in the orange region, the system presents bistability between asynchronous activity state and oscillatory state. There is also a small region (intersection of orange and blue regions) showing bistability between two different types of oscillations. Indeed, in this region the system presents 3 limit cycles, two of them are stable and one is unstable, in addition to the unstable equilibrium point, corresponding to the asynchronous state. Interestingly, we have found that the size of the bistability region depends on several parameters of the model, in particular, the parameters modeling the mutual interpopulations (J_ei, J_ie) and recurrent coupling (J_ee, J_ii). Studying the dynamics in the bistable regions is a topic of interest for understanding the role of oscillations in cognitive tasks. See, for instance, [35] for a study of the response of multistable networks of recurrently coupled spiking neurons to external oscillatory drive and its implications for working memory and memory recall tasks. However, the topic is beyond the scope of this paper and is left for future research.

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Fig 2. E-I cortical network with oscillatory activity in the gamma range.

(A) Two-parameter bifurcation diagram of system (1)-(2) for the excitatory current (x-axis) and the inhibitory current (y-axis). The solid (resp. dashed) blue curve corresponds to a supercritical (resp. subcritical) Hopf bifurcation curve. Red circles correspond to (codimension 2) Generalized Hopf bifurcations (GH), also known as Bautin bifurcations. Purple curve corresponds to saddle node bifurcation of limit cycles. Green circle corresponds to a Cusp bifurcation of limit cycles (CPC), a point where two branches of saddle-node bifurcations of limit cycles meet tangentially. Oscillations occur in the blue and orange regions. Red cross corresponds to values and generating the limit cycle considered later on. (B, C) Frequency oscillation (green) and integral mean values of the firing rates r_e (dashed red) and r_i (dashed blue) as a function of (B) tonic excitatory current and (C) tonic inhibitory current . See Eq (7). (D, E) Time difference (blue) and relative phase (orange) between inhibition and excitation as a function of (D) tonic excitatory current and (E) tonic inhibitory current . In Panels B and D the inhibitory current is set to 0. In Panels C and E the tonic excitatory current is set to 12. Other parameters are as in (6).

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To characterize the oscillations, we compute their frequency and the integral mean values of the firing rate of both populations, that is, (7) where T* is the oscillation period. The values R_e and R_i in (7) provide an average of the firing rate of the individual E and I-cells, respectively.

Moreover, we compute the time difference between the maximum of the I and E firing rates in one cycle (I to E latency) and the relative phase, i.e., the time difference normalized by the period. These factors provide information about the time it takes for the I-cells to activate and halt the activity of the E-cells, and suggest possible gates for communication in the CTC context [25].

We first consider a fixed value of the external inhibitory current (), and we increase the excitatory one from 6 to 15. This interval covers the progression towards the oscillatory region (see solid arrow in Fig 2A). In Fig 2B we show the frequency of the emerging oscillations as a function of the current injected into the E-cells (green curve). Notice that the frequency is bounded away from zero (characteristic of a Hopf bifurcation) and increases from 30 to 70Hz, which fits in the gamma range. The average firing rate of the E and I cells (see Eq (7)) is close to the macroscopic oscillation frequency for all values of the external current that we have explored. Increasing the external input onto the E-cells causes only a slight shortening of the latency period of inhibition over excitation, especially for oscillations above 40Hz (see Fig 2D).

We also vary the constant external drive onto the inhibitory cells from 6 to -2 (notice that it may take negative values), while the input to the E cells is kept fixed at . The system is initially in the non-oscillatory region and enters the oscillatory region (see dashed arrow in Fig 2A). Increasing the depolarizing current onto the I-cells does not significantly affect the macroscopic oscillation frequency (the frequency remains close to 50 Hz, except for the final drop around , see green curve in Fig 2C), in contrast with the effect of a depolarizing current onto the E-cells. However, increasing causes a strong effect on the inhibition-to-excitation latency (compare phase difference with Fig 2D), the inactive period after inhibition acts on excitation shortens, and thus the relative phase (see Fig 2E).

In summary, the oscillation frequency is governed by a trade off between the decay time of the inhibitory conductance and the external excitatory drive onto the E-cells. Thus, increasing the depolarizing drive onto the E-cells () shortens the oscillation period, because excitation can overcome the effect of the inhibition sooner. On the other hand, increasing the depolarizing drive onto the I-cells (), activates the I population faster, thus shortening the I-E latency.

Next, we are going to study the different phase-locking patterns that emerge when the oscillatory neural E-I network receives a periodic external input. To do so, we consider the oscillations that emerge for values and (red cross in Fig 2A). The oscillation frequency is about 41.26Hz (period T* = 24.234 ms), clearly in the gamma range. Moreover, the time difference between the E and I volley peaks is short (around 4ms, see Fig 2D). Fig 3A shows the evolution of the variables of the system along this periodic orbit. Later in Section 2.6, we will explore another type of oscillations, closer to the Hopf bifurcation curve, showing different features.

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Fig 3. Rotation numbers and phase-locking regions for the forced PING oscillation varying input parameters.

(A) Temporal evolution of firing rate, mean membrane potential and synaptic variables over a cycle of a PING oscillation for system (1)-(2) corresponding to the red cross in Fig 2A. (B) Infinitesimal Phase Response Curve (iPRC) for perturbations in the direction of the variables V_e and V_i (red and blue curves, respectively) and the sum of them (purple curve). (C) Von Mises (circular) distribution as a function of the factor κ controlling the input coherence. Large values of κ result in distributions concentrated around the location μ = 0, whereas smaller values lead to broader low-amplitude distributions. The black horizontal line corresponds to the uniform distribution (limit case attained when κ = 0). (D, E) Rotation numbers of the stroboscopic map (9) for a von Mises input (15) applied in the direction of V_e and V_i, as a function of the ratio between the intrinsic period of the E-I network T* and the input period T. (D) Rotation numbers for κ = 2 and different amplitude values A. (E) Rotation numbers for A = 0.1 and different input coherence values κ. (F) Arnold tongues computed using the phase reduction corresponding to the 1:1 (orange), 1:2 (purple) and 2:1 (blue) phase-locked states for different input coherence: κ = 20, 2, 0.5 corresponding to the regions delimited by solid, dashed, dash-dotted curves, respectively. In grey, we show the corresponding Arnold tongues for pulsatile inputs (κ → ∞) obtained analytically. See Methods.

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2.2 Mathematical analysis of phase dynamics

In this Section we describe the mathematical analysis used to study the phase relationship between the neural oscillator described by the low dimensional firing-rate model (1)-(2) and an external time periodic input g(t) (see Eq (5)).

2.3 Increasing the coherence of the periodic input enlarges the frequency range that entrains the network

Beyond varying the input strength and frequency, we have also computed the rotation number and Arnold tongues while varying the input coherence. In this paper, the concept of coherence refers to concentration of the periodic input volley around some phases of the cycle (as in [22]). The input coherence is modulated by the parameter κ in the input Eq (15), while the total external drive over a cycle is maintained, that is, the area under the input is independent of κ (see Fig 3C and Methods). Moreover, we also consider the limit case (κ → ∞), the input then becomes a periodic pulse and the pulses are modeled by delta functions. In this case, the Arnold tongues can be computed analytically. See Methods.

We observe that higher coherent inputs (larger κ) give rise to devil’s staircases with larger rational plateaus (compare the red curves with the blue ones in Fig 3E). Moreover, when looking at the Arnold tongues of the largest phase-locking regions in the devil’s staircases (1:1, 1:2 and 2:1), we observe that lower coherent inputs (small κ) result in narrower Arnold tongues, enclosed in those with larger κ (see Fig 3F). Thus, inputs with higher coherence are capable of entraining the target network for a wider range of frequency values. Noticeably, as input coherence increases, the right boundary of the Arnold tongues is set approximately at a fixed frequency relationship T/T* ≈ p/q independently of the amplitude A (notice the right boundary of the 1:1 and 1:2 Arnold tongues for κ → ∞ is fairly vertical). Thus, increasing the input coherence enlarges the range of frequencies that entrain the network without shifting it.

In the following Section, we explore whether the type of entrainment that emerges inside the 1:1, 1:2 and 2:1 phase-locking regions for coherent inputs (κ = 2) is optimal for communication in the CTC context.

2.4 Inputs with higher frequency communicate more effectively

In this Section, we explore the implications of phase-locked states for neuronal communication in the CTC context. In the previous Section, we have identified several phase-locked states between an oscillating E-I network (PING interplay) and a periodic input using the phase reduction (see Fig 3). However, phase-locking is not enough to guarantee effective communication; the phase relationship must satisfy certain conditions. The CTC theory proposes that the phase difference between oscillating neuronal groups is adequately set up for communication if windows for sending and receiving are open at the same time, because then the inputs from the pre-synaptic group will produce changes in the firing rate of the post-synaptic population and communication will occur [8, 9].

To identify how information can be encoded in the input, transmitted and represented in the output of the firing rate of the E-cells, we consider periodic inputs with different amplitudes modulated by the parameter A (when A = 0, the stimulus is not present) and we measure the changes in the firing rate of the post-synpatic group, either on the average firing rate or in the spike synchronization, as suggested in [21]. If inputs with different amplitudes generate different output responses, then the information encoded in the amplitude of the input will be detected at the output and communication will occur.

Moreover, CTC theory proposes that effective communication depends essentially on the timing of the inhibition and the external input [37]. In the unidirectional setting considered herein (see Fig 1A), if the external input volley reaches the receiving E-population before the inhibition activates, it may elicit a response of the E-cells. Thus the target network will detect changes in the input strength by means of its firing rate and render communication effective. On the other hand, if the input volley arrives to the E-population when the inhibition is present, then such an input may have no effect on the activity of the E-cells and information will not be transmitted to the target network.

In this Section, we provide four measures that will allow us to characterize quantitatively whether the input strength A is transmitted to the receiving population, how it is represented by the activity of the E-cells of the target network and how this transmission is related to the phase-locking properties of the entrainment. In particular, we compute the factor Δτ (Eq (23)), which measures the time difference between the peaks of the input and the I-volley, and the factors , Δα and Δσ which measure the firing rate response of the E-cells of target network to the input (see Methods). More precisely, (Eq (24)) provides information about the rate change in the overall firing rate of the E-population over the whole cycle, Δα (Eq (25)) about the rate change in the maximum of the E-volley and Δσ (Eq (26)) about the rate change in the half-width of the E-volley. In our previous work [24], we considered only the factor Δα, but the quantities and Δσ allow us to complement the information about the changes in the firing rate. Thus, the factor Δτ describes the properties of the phase-locking that emerges, in particular, how close in time is the input from the activation of the I-cells, while the other factors describe how the network firing rate is affected by the changes in the input strength.

In Fig 4 we show the factors Δτ, , Δα, and Δσ for periodic solutions of system (1)-(2) perturbed with a von Mises input Ap(t) with coherence κ = 2 (see Eq (15)), in the 1:1 phase-locking region (identified in Fig 3), where the existence of these orbits is guaranteed. The factors were computed along constant amplitude values that range between A = 0.01 and A = 0.2 in the corresponding Arnold tongue.

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Fig 4. Input effects on the E-cell evoked response for a network entrained by coherent inputs.

(A) Evolution of the firing rate variables r_e (red) and r_i (blue) of the perturbed system (1)-(2) with a von Mises input with κ = 2 (dashed green) for a representative periodic orbit within the 1:1 phase-locking region. (B-E) Factors describing changes in the E-cell response within this phase-locking region for orbits of the perturbed system (1)-(2) calculated along (equidistant) sections A = ct of the corresponding Arnold tongue, indicated by the color of the curve (ranging from dark blue, A = 0.01, to yellow, A = 0.2, with increments of size 0.01). The factors are: (B) Δτ, describing the timing between inhibition and input volleys (normalized by the input period T), (C) , describing the rate change in the averaged firing rate of the E cells, (D) Δα, describing the rate change in the maximum of the firing rate of the E cells, and (E) Δσ, describing the rate change in E-volley half-width. See Methods.

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The entrainment presents different features inside the 1:1 phase-locking region depending on the frequency of the external input. The factor Δτ (see Fig 4A) indicates that when the input frequency is close to the natural gamma cycle (T/T* ≈ 1), the input volley reaches the E-population when the I-cells are active (Δτ ≈ 0 or even smaller) and, as the input frequency increases (T/T* decreases away from 1), the time by which the input volley precedes the I-population volley increases (Δτ increases up to 0.4). See Fig 4B.

The factor (Fig 4C) indicates that the increase in the mean firing rate of the E-cells due to changes in the input amplitude A is stronger for high frequency inputs (i.e. T/T* close to the left boundary) than for inputs with frequencies similar to the natural gamma cycle (i.e. T/T* close to the right boundary). However, in all cases we observe a weak modulation of the mean firing rate activity by the input strength A (values of below 1.3).

These differences between input frequencies in the modulation of the output response due to changes in the input strength A are more noticeable when looking at the factors Δα and Δσ, describing the shape of the E-volley, that is, the spiking synchronization properties within the target network. For higher input frequencies there is a significant sharpening of the E-volley and therefore an increase in the spike synchronization of the post-synaptic group when A increases: the half-width decreases (Δσ much smaller than 1) and the maximum increases (Δα much greater than 1) (see Fig 4D and 4E). For frequencies T/T* ≈ 1 (close to the right boundary), the maximum and half-width of the E-volley barely changes when A increases (Δα ≈ 1 and Δσ ≈ 1). Indeed, it slightly widens (i.e. the E-cells become less coherent) because the input arrives close to the natural activation of the E-cells. We will see later in Section 2.6 that this effect is more pronounced in the case of oscillations close to the Hopf bifurcation.

Therefore, we have found that changes in the input strength A are transmitted and better reproduced at the output by the spike synchronization properties of the E-population (described by both the maximum firing rate and half-width of the E-volley) rather than by the average firing rate r_e. Moreover, they are better detected for periodic inputs with higher frequency than the natural gamma cycle. Thus, we establish that communication between input and output is quantified by means of the factor Δα (and, also the factor Δσ), which measures the amplification of the oscillations of the target network (the receiver) due to the periodic perturbation. Thus, we conclude that our results suggest that the entrainment of the network by higher input frequencies than the natural gamma cycle may result in a more effective communication of the information encoded in the input strength.

We acknowledge that we have studied an E-I network driven by a periodic input in the absence of noise, thus our setting is deterministic. However, our predictions on effective communication might be tested in the presence of noise using techniques from information theory. We have run preliminary computations that we include in the S1 Appendix, that seem to agree with our deterministic observations. Further explorations are left for future work. See the Discussion.

We have also explored the type of entrainment that emerges in the 1:2 and 2:1 phase-locking regions (see S3 Fig). In the 1:2 phase-locking region, the network receives two input volleys but responds with only one E-I volley every cycle (see S4 Fig). Thus, only one input volley elicits a response in the target network, while the other one is ignored, due to the activation of inhibition. We observe that the E-volley is modulated in a similar way as in the 1:1 phase-locking region (see panels C-E in S3 Fig).

Finally, in the 2:1 phase-locking region, we find two E/I-volleys per one of the input (see S5 Fig). Thus, the input affects mainly only one E/I-volley (solid line in S3 Fig right), while the second E/I-volley reflects only the intrinsic dynamics of the network and it is slightly modulated by the external input. Interestingly, when the input frequency is decreased, the input modulates both E/I-volleys. Then, in these cases, the low frequency input can be seen as a modulator of the E/I-network rhythm (see Discussion).

2.5 Inputs in competition and selective communication

Thus far we have studied how a single periodic external input may entrain the network and how is the communication established between the emitting and receiving populations. In this Section we want to explore how CTC theory implements selective communication. In selective communication the target network receives several stimuli from different sources, but responds only to one of them (the relevant stimulus), while ignoring the others [9].

To identify the relevant properties that are involved in selective communication, we probe how the phase-locked states described in the previous Section are affected by the presence of a distractor. In particular, we consider the case where the target network is entrained by a primary periodic input A₁p₁(t) of the form (15) with frequency f₁ = 1/T₁ and coherence κ₁ = 2 (coding for the attended stimulus) and we add a second periodic perturbation A₂p₂(t) (coding for the distractor) of the form (15) for which we will vary the frequency f₂ = 1/T₂ and the coherence κ₂ (see Fig 5A). We investigate first whether the distractor prevents or not the target oscillatory network to follow the rhythm of the primary stimulus. Second, we investigate how changes in the amplitude of the primary and the distractor affect the output firing rate to determine how the information stored in the amplitude of the primary is transmitted and detected at the output by means of the E-cell evoked response. This approach allows us to identify whether there is a stimulus being detected (attended stimulus) and another ignored and the underlying mechanisms for this selection.

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Fig 5. Effect of an identical frequency distractor in the network entrained by the primary stimulus.

(A) Schematic representation of an E-I cortical neural network (PING interplay) receiving two oscillatory inputs from different sources: the primary input A₁p₁(t) (green circle) and the distractor one A₂p₂(t) (red circle). (B) Rotation numbers ρ of the stroboscopic map (9) for a perturbation consisting of a primary input and a distractor. Both inputs are modeled by means of a von Mises distribution, have the same amplitude factor A₁ = A₂ = 0.1 and the same period T = T₁ = T₂ but phase-shifted. The coherence for the primary is fixed at κ₁ = 2. We vary the distractor coherence κ₂ (x-axis) and the period T, so that the values T/T* (color legend) are distributed along the 1:1 plateau for κ₁ = 2 (the oscillator and the primary stimulus support a 1:1 phase-locking relationship). If ρ = 1, the entrainment by the primary stimulus is preserved despite the presence of the distractor, otherwise, it breaks down.

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2.5.2 Two input streams of different frequency.

We now turn to study the effect of a distractor of different frequency than the primary that entrains the network. We fix the parameters κ₁ and T₁ of the primary input A₁p₁(t) so that there is a 1:1 phase-locking relationship between the primary and the oscillatory E-I network. Then, we add a second external input A₂p₂(t), for which we vary the coherence κ₂ (between 0.01 and 20) and the period T₂ (within the range ). The distractor is phase-shifted so that both input volleys are as much apart as possible over a cycle (see Fig 6A and 6B and Methods for a precise definition). The amplitude for both inputs is set to A₁ = A₂ = A = 0.1.

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Fig 6. Effect of a non-identical distractor in the network entrained by the primary stimulus.

(A, B) Two von Mises inputs of different frequency phase-shifted according to the rule described in Methods. The black curve corresponds to the primary stimulus located about μ₁ = 0 with κ₁ = 2 and period T₁ = T. The other curves correspond to a distractor with different coherence values κ₂ indicated in the legend. The distractor is phase-shifted (A) T₂/2 if T₂/T₁ < 1 or (B) T₁/2, otherwise. (C) Arnold tongue corresponding to 1:1 phase-locking between a single von Mises input with coherence κ₁ = 2 (primary input) and the target network. We have selected 3 orbits along the section A₁ = A = 0.1 (black crosses) corresponding to T₁/T* = 0.845, 0.93 and 1 (left to right), for which we apply a distractor input of the same strength A₂ = A = 0.1. (D, E, F) Synchronization index r for the stroboscopic map at time T₁ as a function of the coherence of the distractor κ₂ (x-axis) for different values of the periods ratio between inputs, T₂/T₁ (color legend). The distractor frequency can be higher (cold colors) or lower (warm colors) than the primary frequency, being as much twice as fast (dark blue line) or 3/2 times slower (red line).

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Notice that in this case the perturbed system (1)-(2) with g(t) = A₁p₁(t) + A₂p₂(t) is no longer T₁-periodic, neither the phase Eq (8). However, we still consider the time-T₁ map of the phase equation, which provides a phase over the cycle at each time T₁ (although the evolution of this map depends also on time). See Methods for more details. We compute the synchronization index (SI) r, also known as vector strength, for the phases of this map. The synchronization index is a measure of how clustered are the phases on a cycle [38, 39]. Thus, if r ≈ 1, the phases are all clustered around the same value, while if r ≈ 0 they are scattered around the circle. See S6 Fig.

The frequency relation T₁/T* has been fixed at three different points inside the 1:1 Arnold tongue for κ₁ = 2 along the line A₁ = A = 0.1: on the left (T₁/T* = 0.845), in the middle (T₁/T* = 0.93) and on the right-hand side (T₁/T* = 1). See Fig 6C. In Fig 6D-F we show the SI as a function of the frequency relation between the primary input and the distractor (f₁/f₂ = T₂/T₁) and the coherence of the distractor κ₂. We consider that the entrainment of the target network by the primary input is maintained despite the presence of the distractor when SI satisfies r > 0.8.

We observe that the entrainment is not disrupted by the presence of a distractor when the relation T₁/T* lies near the leftmost edge of the 1:1 Arnold tongue (see Fig 6D for T₁/T* = 0.845). The distractor can either oscillate faster or slower (within the range ) or it might even come in the form of sharper or broader pulses (vary κ₂), but it does not prevent the primary input to entrain the oscillator. Notice that the SI remains above 0.8 for all values of κ₂ and frequency relations T₂/T₁.

When T₁/T* lies in the center of the 1:1 Arnold tongue, faster distractors than the primary input (i.e. T₂/T₁ ≤ 1) cause the phase-locking breakdown since the SI falls below 0.8 (see cold color curves in Fig 6E), independently of the distractor coherence κ₂, as long as it is above 1. However, slower distractors than the primary input (i.e. T₂/T₁ ≥ 1) do not disrupt phase-locking significantly. Indeed, the SI remains above 0.8 (see warm color curves in Fig 6E).

Finally, when T₁/T* lies close to the right border of the 1:1 Arnold tongue, the distractor succeeds in completely disrupting the 1:1 phase-locking, irrespective of κ₂ and the frequency relationship T₂/T₁ (see Fig 6F). Indeed, the SI for all values is below 0.5.

In conclusion, if the distractor is coherent enough, what determines the robustness of the entrainment is the frequency relationship T₁/T*. Indeed, the entrainment of the network by the primary input is much stronger (robust to distractors) when the primary has a higher frequency than the natural gamma cycle. In this case, even the more coherent distractors cannot break the 1:1 phase-locking. As the primary input frequency decreases and approaches the natural gamma cycle frequency, the entrainment becomes weaker because, first, faster distractors can break the synchrony and finally, close to the right boundary, any distractor can do so.

In the next Section we will discuss the underlying mechanisms that explain the dependency of the entrainment robustness on T₁/T*.

2.5.4 Effects of the disruptor on the E-cell evoked response.

We are not only interested in studying whether entrainment is preserved but also how the communication (i.e., how the input strength of the primary input affects the E-cell firing rate respones) is affected by the presence of a distractor. More precisely, in those cases in which the entrainment by the primary input is maintained, we want to explore whether the presence of a distractor affects the response of the E-cells of the target network, and therefore disrupts the communication with the primary. A question that arises is whether the output of the E-I network reflects summation in some sense of the two inputs or whether one dominates over the other.

Exploring in detail the influence of a distractor in the output firing rate and, in consequence, the communication, involves several parameters, such as the relative frequency T₁/T₂ and relative amplitudes A₂/A₁ between both inputs (primary and distractor), together with the relative frequency with respect to the natural gamma cycle. In this Section, we highlight the qualitative role played by the most important parameters and we leave the systematic study for future research.

The main findings are summarized in Fig 7, where we show for some representative examples, the firing rate r_e of the E-cells in the absence of input (yellow dashed curve), when only the primary input is present (orange dash-dotted curve) and when both the primary and the distractor inputs are present (red solid curve). The representative examples chosen are with T₁/T* as in Fig 6 and T₂/T₁ = 1.2, κ₁ = κ₂ = 2.

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Fig 7. Effects of the disruptor on the E-cell evoked response for some representative trails.

Three different types of entrainment inside 1:1 phase-locking region corresponding to (A) T₁/T* = 0.845, (B) T₁/T* = 0.93 and (C) T₁/T* = 1. (Top) Time evolution of the excitatory firing rate r_e for system (1)-(2) in the absence of perturbation (dashed yellow curve), when the network is entrained by the primary input (dashed-dotted orange curve) and when both the primary input and the distractor are present (solid red curve) for a time duration of 6T₁. (Bottom) The primary input (solid black) and the distractor (dashed black) correspond to inputs of von Mises type with A₁ = A₂ = A = 0.1, κ₁ = κ₂ = 2 and a frequency relationship T₂/T₁ = 1.2. (D, E) Factors Δα, Δσ and describing changes in the E-cell evoked response for the case T₁/T* = 0.845 (primary on the left-hand side of the Arnold tongue) and T₂/T* = 1.2T₁/T* = 1.04 (distractor on the right-hand side of the Arnold tongue) when (D) the amplitude of the primary A₁ is varied from 0 to 0.2 while keeping the amplitude of the distractor fixed at A₂ = 0.1 and (E) viceversa. We have computed the mean and standard deviation of these factors over 10 cycles of the primary input.

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For values of T₁/T* on the left-hand side of the 1:1 Arnold tongue, the I-volley is in anti-phase with the primary input volley. Thus, the windows for the disruptor to evoke a response are only open during the arrival times of the primary volley. Therefore, depending on the frequency relationship T₂/T₁, we have that, for some cycles, the input volley of the distractor and the primary input almost coincide, thus providing more concentrated input drive to the E-cells and evoking a higher response, while for other cycles, the distractor input volley reaches the E-cells when inhibition is present and the network does not respond to the distractor. See also S7 Fig left and panel A in S8 Fig.

For values of T₁/T* in the middle region of the Arnold tongue, the presence of a distractor decreases on average the evoked response of the E-cells to the primary input, even when the network is still entrained by the primary (compare red solid and orange dash-dotted curves in Fig 7B). The reason is again related to the timing of the I-volley. The time difference between the primary input volley and the I-volley has shortened (see Fig 4B), thus offering more temporal windows (away from the primary input volley) for the disruptor to elicit a response on the E-cells. This causes that for certain cycles the disruptor volley elicits a response ahead of the primary, thus affecting the overall response. See also S7 Fig middle and panel B in S8 Fig.

For the case T₁/T* = 1, the primary input elicits a weak response on the E-cells (compare yellow dashed and orange dash-dotted curves in Fig 7C), while the distractor increases the response in the E-cells (compare red solid and orange dash-dotted curves in Fig 7C). In this case, most of the distractor volleys arrive when inhibition is low, thus causing the activation of the E-cells and affecting the entrainment. See also S7 Fig right and panel C in S8 Fig. In this case, the output firing rate reflects the effects of both primary and distractor inputs.

Next, we take a more systematic approach to assess the contribution of each input to the E-cell evoked response in a particular setting. We consider the case where the primary input has a frequency relationship with the natural gamma cycle that lies on the left-hand side of the Arnold tongue (T₁/T* = 0.845), and we consider a distractor that has a frequency T₁/T₂ = 1.2 (T₂/T* = 1.04, thus, it lies on the right-hand side of the Arnold tongue). Recall that we have seen in Section 2.4 that inputs with higher frequency communicate more effectively, so we will explore whether this is maintained when we add the distractor. We computed the average changes in the maximum firing rate of the E-cells Δα, half-width of the E-volley Δσ and average firing rate over several cycles in the presence of a distractor with strength A₂ = 0.1 and for varying A₁ (see Fig 7D). We repeated the computations fixing the strength of the primary (A₁ = 0.1) and varying A₂ (see Fig 7E). We can observe that increasing A₁ from 0 (stimulus off) to 0.1 increases significantly the firing rate synchronization within the E-population (described by an increase in Δα and a decrease in Δσ) in the presence of a the distractor (Fig 7D), suggesting that information about stimulus state described by the input strength might be efficiently transmitted. However, increasing the strength of the distractor when the primary input is present affects very little the firing rate of the E-cells both in terms of average firing rate and sharpening of the E-volley (very small variation of the factors Δα, and Δσ in Fig 7E), suggesting that this input is ignored. Remarkably, in all cases, the average firing rate of the E-cells over a cycle shows little modulation by the strength of the primary and distractor, thus producing little encoding.

This result suggests that the network is capable to detect changes in the strength of a target periodic input, while ignoring simultaneously active distracting periodic inputs, if the frequency of this input is higher than the natural gamma cycle. Again these predictions might be tested in the presence of noise using tools from information theory (see supporting information S1 Appendix for preliminary results and the Discussion).

2.5.5 Selective communication and switching between identical input streams.

When the E-I network receives two identical input streams, we have already identified a range of frequencies, for which the target network oscillates coherently (phase-locked) with both of them. Interestingly, the system presents bistability between two possible phase relationships. We want to explore whether these two phase relationships correspond to the network responding differently to each input, suggesting selective communication. Finally, we will study how the network may switch between these two possible phase relationships, thus suggesting a mechanism for switching between attended inputs [16].

Consider an oscillatory E-I neural network entrained by two identical input streams A_i p_i(t), i ∈ {1, 2} of the form (15) with A_i = 0.1 located in anti-phase (see, for instance, Fig 8A and panel C in S7 Fig left). This situation occurs in the 1:2 Arnold tongue (see Fig 3F and S4 Fig), and similarly, in the case of two identical inputs with a particular period relationship with the natural gamma cycle T/T* (those values such that the rotation number ρ = 1 in Fig 5B). Notice that in the latter case, the 1:1 phase-locking between the target network and the first input is not disrupted by the addition of a secondary one.

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Fig 8. Selective communication and switching between attended stimulus.

(A) Time evolution of the excitatory and inhibitory firing rate r_e (red), r_i (blue), respectively for system (1)-(2) receiving two identical inputs of von Mises type in antiphase (κ_1,2 = 2 and T = T_1,2 = 0.84T*, A_1,2 = 0.1). We establish that the closest input volley to the E-volley corresponds to the primary input (black solid curve), and the other one to the distractor (black dashed curve). (B, C) Factors Δα, Δσ and describing changes in the E-cell evoked response (B) when the amplitude of the primary A₁ is varied from 0.1 to 0 while keeping the amplitude of the distractor fixed at A₂ = 0.1 and (C) viceversa. We have computed the mean and standard deviation of these factors over 10 cycles of the primary input. (D) Phase Response Curve (solid blue) obtained by applying square-wave perturbations of amplitude 1.5 and duration 2 ms at different phases of the periodic solution in panel A. The plot also shows if the (entrained) oscillator remains in the same periodic solution before and after the pulse administration (black solid line) or if the oscillator switches to a different solution where the roles of the primary and distractor are exchanged (dashed black line). (E, F) Simulations of the full spiking QIF model showing the response of the network to a square-wave current delivered at two different phases of the cycle: (E) t/T = 0.3, for which no switching between attended stimuli occurs, and (F) t/T = 0.5, for which switching occurs. Each panel shows (from top to bottom), for a time interval of 150 ms, the two identical von Mises inputs in antiphase (solid and dashed black) with the mean firing rates of the E-cells (red) and I-cells (blue) of the full spiking QIF model, the corresponding raster plot and the time at which the square-wave pulse is applied. We have integrated the full network of QIF neurons for 1000 ms. At time 200 ms we apply the two inputs of von Mises type. The square-wave pulse is applied at time 200 + 23T + t ms.

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Given a solution corresponding the network being entrained at a particular phase relationship by two identical inputs, we establish that the closest input volley in time to the E-volley corresponds to the primary input and the other one to the secondary (see Fig 8A). We will explore what is the contribution of each input to the output firing rate, by means of modifying the input strength. We vary the strength of the primary input A₁ away from 0.1, while keeping the distractor fixed (A₂ = 0.1). We observe that the synchronization of the E-cells decreases when A₁ decreases from 0.1 (Δα decreases and Δσ increases, see Fig 8B), until the primary input is too weak and the distractor takes over (this corresponds to values of A₁ ≈ 0.06 in Fig 8B). However, when we repeat the same process but keeping the primary input fixed at A₁ = 0.1 and varying the amplitude of the distractor, we observe that the response of the E-cells shows the opposite trend. That is, decreasing the strength of the distractor increases the synchronization properties of the E-cells (Δα increases and the Δσ decreases, see Fig 8C). Based on this observation, we interpret that the network responds to the primary, because the primary amplifies the oscillations of the target network, while the distractor interferes in the communication of the primary (reduces the amplification effect of the primary). For the other phase relationship, the problem is symmetrical and the role of both inputs is reversed. The mean firing rate barely changes in both cases (see in Fig 8B and 8C).

We study the effects of a brief/short stimulus (a pulse) to the entrained network. Brief stimuli could transiently lengthen or shorten the cycle period (i.e. phase shifting) and force the oscillator to switch phase-locking, thus producing a change in the role played by the inputs. To quantify these effects, we consider the periodic solution for the perturbed 8-dimensional model (1)-(2) with g(t) = A₁p₁(t) + A₂p₂(t) and apply a square wave pulse of amplitude 1.5 and duration 2 ms (i.e. brief and short stimulus) to the E-cells. In Fig 8 we show, for a particular case, the change in phase (blue curve) and in the effective input (black solid/dashed line), as a function of the phase t/T of the entrained oscillator at which the pulse is applied. See Methods. Solid line denotes no change in effective input—the oscillator responds to the same input stream after the pulse—and dashed line denotes a change—the oscillator responds to the other input stream after the pulse. Note that the phase shift is positive almost for all phases, meaning that a pulse typically causes a phase advance (i.e. the E-volley occurs earlier). Notice that there is a small range of phases (between 0.4 and 0.8 approximately) for which a square wave pulse produces a switch in the effective input. This interval corresponds to phases of the cycle for which the I-volley just passed and the next E-volley has not arrived yet (see, for instance, Fig 8F). Notice that the phase shift is close to zero if the pulse arrives at the same time as the I-volley (see, for instance, Fig 8E).

We simulate the full microscopic model to support and illustrate these results. A pulse perturbation of the full microscopic model at phase 0.3 does not entail a change in the effective input (oscillator still responds to the dash-lined input after the pulse is applied in Fig 8E), while at phase 0.5 it does cause a change in the effective input (oscillator responds to the solid-lined input after the pulse is applied in Fig 8F), as predicted by the reduced model.

2.6 Oscillations close to the Hopf bifurcation

Thus far we have observed that the timing and shape of the I-volley in the gamma cycle plays a relevant role in implementing several aspects of the CTC theory. To emphasize better this role, we explore how CTC theory implements in a different type of gamma cycle. More precisely, we consider oscillations arising from the PING mechanism but closer to the Hopf bifurcation curve in the parameter space (see Fig 2A). Thus, for parameters as in (6) and tonic currents to the E and I-cells , , respectively, the system (1)-(2) presents oscillations at a frequency 1/T* = 1/30.5 ≈ 32.78Hz (lower gamma range) of the form shown in Fig 9A. Notice that E/I-volleys are wider and lower for this oscillation than the ones considered along this paper with (compare with Fig 3A).

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Fig 9. PING oscillations close to a Hopf bifurcation.

(A) Temporal evolution of firing rate, mean membrane potential and synaptic variables over a cycle of a PING oscillation for system (1)-(2) corresponding to external current (close to the Hopf bifurcation curve in Fig 2A). (B) Infinitesimal Phase Response Curve (iPRC) of the cycle in Panel A for perturbations in the direction of the variables V_e and V_i (red and blue curves, respectively) and to both of them (purple curve). Note that the iPRC is both positive and negative. (C) Rotation numbers of the stroboscopic map (9) for a von Mises input (15) with coherence κ = 2 applied in the direction of V_e and V_i, as a function of the ratio between the intrinsic period of the E-I network T* and the input period T and different amplitude values A. (D, E, F) Time evolution of the mean firing rates of the E-cells (red) and I-cells (blue) along the unperturbed (dashed curves) periodic orbit for the system (1)-(2) and perturbed (solid curves) with the coherent von Mises input with A = 0.05 and relative frequency T/T*: (D) 0.9, (E) 1 and (F) 1.07. The period of the oscillators has been normalized to 1.

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Moreover, we also compute the iPRC for this oscillator (see Fig 9B). The iPRC takes both positive and negative values, as opposed to the oscillator with (compare with Fig 3B). This means that this oscillator may be entrained by either faster or slower periodic inputs. This is the typical behavior for oscillations close to a Hopf bifurcation [31].

We perturbed this oscillator with a periodic input g(t) = Ap(t), with p(t) of von Mises type with κ = 2 (see Eq (15)) and applied the phase reduction method described in Section 2.2. The rotation numbers for the stroboscopic map (9) as a function of the relative frequency (T/T*) are plotted in Fig 9C for different amplitude values. The visible plateaus, corresponding to rational rotation numbers (ρ = p/q), are centered about the point T/T* = p/q and extend towards both sides. Therefore, the oscillator can be entrained by faster (i.e. T/T* < p/q) and slower inputs (i.e. T/T* > p/q), as expected from the iPRC shape. Notice that the largest plateaus correspond to 1:1 and 2:1 phase-locking (ρ = 1 and ρ = 2, respectively) and they grow as the amplitude A increases.

Since the rotation numbers in Fig 9C have been computed using the phase reduction approach, the phase-locking predictions may not be accurate for the perturbed 8-dimensional mean-field model (1)-(2), specially when the amplitude increases. Indeed, we observe that the actual phase-locked solutions do not extend as far to the right (towards lower input frequencies), rather they extend a little to the left (towards higher input frequencies) than predicted by the phase reduction. Thus, for A = 0.05, the phase reduction identifies 1:1 phase-locking for T/T* ∈ [0.883, 1.11], but the actual computation finds synchronized solutions for T/T* ∈ [0.861, 1.07].

Even if 1:1 phase-locking for low input frequencies does not extend as far to the right as predicted, we still find entrained solutions for inputs with frequencies lower than the network intrinsic frequency (T/T* > 1). It is interesting to explore how is the response in the target network for low input frequencies, compared to those with high input frequencies. We look for periodic solutions of the perturbed mean-field model (1)-(2) along the 1:1 phase-locking plateau for A = 0.05. For T/T* = 0.9 (high input frequency) the entrained oscillator shows a significant increase of the firing rate activity of both E and I-cells (compare solid and dashed lines in Fig 9D). As the frequency of the input becomes more similar to the natural gamma cycle, the E-volley becomes lower and wider (see Fig 9E and 9F for T/T* = 1 and T/T* = 1.07, respectively). This is because the E-cells activate due to both, the intrinsic properties of the network and the input volley, the latter arriving shortly after the E-cells start to activate but before they can trigger inhibition. Thus, the inputs with low frequencies contribute to widen the E-volley, but without increasing its maximum significantly (the E-cells become less coherent). For larger values of T/T* we do not find periodic solutions anymore.

To better identify the aspects related to CTC theory, we compute the factors Δτ and (as well as Δα and Δσ) introduced in Section 2.4 to measure the timing of the inhibition and the changes in the excitatory firing rate due to the perturbation, respectively. We have computed such magnitudes for the solutions in the 1:1 phase-locking region for amplitudes A = 0.025, A = 0.05, A = 0.075 and A = 0.1 in Fig 10). The factor Δτ indicates that the input volley precedes inhibition (0 < Δτ < 0.5) for frequencies higher than the natural gamma cycle and decreases as the input frequency decreases (T/T* increases) indicating that the input volley overlaps with inhibition when the input frequency is close to the natural gamma cycle (Δτ ≈ 1 and below 1) (see Fig 10A). Accordingly, the factor measuring the increase in the overall firing rate also reduces as T/T* increases up to 1 and only for the largest amplitudes (see Fig 10B), and the E-volley becomes wider (Δα decreases, see Fig 10C, and Δσ increases, see Fig 10D). Interestingly, for values T/T* above 1, the average firing rate increases again (Fig 10B) because the E-volley becomes wider (Δσ is above 1), thus causing a decrease in the firing synchronization within the E-population.

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Fig 10. Input effects on the E-cell evoked response for entrained network close to a Hopf bifurcation.

Factors (A) Δτ, (B) , (C) Δα and (D) Δσ describing changes in the E-cell response within the 1:1 phase-locking region for orbits calculated along the amplitudes A = 0.025 (dark blue curves), A = 0.05 (blue curves), A = 0.075 (green curves) and A = 0.1 (yellow curves). Factors are computed for periodic solutions of the system (1)-(2) perturbed by a von Mises type of input with κ = 2. Notice that the periodic solutions are found for smaller intervals than those predicted by the rotation number using the phase reduction (see Fig 9C).

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4.1 Neuron models

We consider a population of N neurons with all-to-all coupling, subdivided into a population of N_e excitatory and N_i inhibitory cells, whose individual voltage dynamics are modeled by a Quadratic Integrate-and-Fire (QIF) model [59], i.e., (10) where V_j is the voltage of the j-th neuron, τ is the time constant, η_j is a constant bias current and I_j is an input current accounting for external (time-dependent) stimuli and the excitatory/inhibitory synapses of the network. The threshold and reset voltages are taken V_th = −V_reset → ∞, (in numerical simulations, we will set them to V_th = −V_reset = 500). Heterogeneity in the system is introduced by assuming that the parameters η_j are distributed according to a Lorentzian distribution with half-width Δ and centered at , (11)

The input current I_j consists of a common external input for all neurons and synaptic current due to recurrent connexions in the circuit. For the excitatory ensemble, the input is expressed as (12) while for the inhibitory one as (13)

We have omitted the subindex j (labeling each neuron in the population) because all the neurons in the same population receive the same input current. The subscript indicates whether it corresponds to the excitatory (E) or the inhibitory (I) population. The variable S_ab models the synaptic current from the pre-synaptic population b to the post-synaptic population a, where a, b ∈ {e, i}. The dynamics of the synaptic currents are described by a linear differential equation of the form (14) where a, b ∈ {e, i}. Here, is the synaptic time constant of population b, J_ab the synaptic strength from population b to population a, and r_b the firing rate of population b, which is computed as where δ(t) is the Dirac delta function and are the firing times of neuron n.

In [26, 29], one can find the details of the derivation of the reduced mean field model (1)-(2) that provides an exact description (in the thermodynamic limit) of the macroscopic quantities of the spiking model described above, namely, the mean firing rate r_a and membrane potential V_a.

4.2 Inputs

To model the excitatory input drive to the E and I cells, and , respectively (see Eq (5)), we consider a periodic function g(t) = Ap(t), for which we can modulate its coherence, that is, how synchronized are the spikes of the pre-synaptic population providing periodic input. To do so, we use the von Mises probability density function (also known as the circular normal distribution) and define the T-periodic function p(t) as (15) where I₀(x) is the modified Bessel function of order 0. The parameter μ is the mean and κ is the temporal coherence (κ = 0 corresponds to a uniform distribution and as κ increases the distribution becomes more concentrated about the angle μ. In the limit case, κ → ∞, the distribution becomes a delta function). See Fig 3C. The parameter α is chosen as α = T so that the temporal average over one period is 1. That is, (16)

When only one input is present, we fix μ = 0 and we vary the coherence κ and the period T.

We also consider the case when the E and I populations receive two input streams of the form (15), one referred to as the primary stimulus and denoted by A₁p₁(t), and the other as the distractor and denoted by A₂p₂(t). The mean, temporal coherence and period of the i-th input will be denoted by μ_i, κ_i and T_i, respectively.

When both inputs have the same period (T₁ = T₂ = T), we place the second input to be in antiphase with respect to the first one, that is μ₂ = T/2. However, when the periods are different, the last criterion is not well-defined. Therefore, we implement the following rule: we place the peak of the input with largest period between two consecutive peaks of the input with shortest period. That is, if T₂ ≥ T₁, then , whereas if T₁ > T₂, .

4.3 Phase reduction

For an oscillating E-I network, the 8-dimensional system (1)-(2) has a hyperbolic attracting limit cycle Γ of period T*. The limit cycle can be parametrized by the phase variable θ, such that θ(t) = t + θ₀(mod T*) and there exists a function such that parametrizes the periodic orbit Γ.

We apply an external periodic input g(t) to the limit cycle Γ in the direction given by the vector . If the external periodic drive g(t) is weak, i.e. |g| ≪ 1, we can reduce the study of the dynamics of the perturbed system to a single equation for the phase variable, given by (17) where the function is the infinitesimal Phase Response Curve (iPRC). The iPRC measures the oscillator’s phase shift due to an infinitesimal perturbation applied at different phases of the cycle [31]. Notice that Z(θ) is a vector of 8 components, thus, the i-th component (Z_i) corresponds to the phase shift due to an infinitesimal perturbation applied in the direction of the i-th variable (i.e. ). It is well known that the iPRC is the periodic solution of the so-called Adjoint Equation [31], given by (18) subject to the normalization condition (19) where the matrix M(γ(t)) is the linearization of the system (1)-(2) around the limit cycle Γ.

4.4 Rotation number

The existence of periodic points for the stroboscopic map (9) defined on the circle is given by a well known result in the context of circle maps [36, 60]. Let be a map on the circle, orientation preserving and regular enough, then the lift of f is a continuous function that satisfies , where π(x) = x mod T* (i.e. the phase on the circle). Then, the rotation number ρ of the map f is defined as (20) for any . Importantly, the limit above always exists and does not depend on the initial point x, nor the lift chosen.

There are two important results to characterize the dynamics of the map f using the rotation number. If ρ is rational (ρ = p/q with ), then there exists a q-periodic point of the map f, that is, a solution of f^q(θ) = θ mod T*. If, by contrast, ρ is irrational, then the orbits of f fill densely the circle and there are no periodic points.

We use the numerical procedure described in [61] to compute the rotation number of the stroboscopic map (9) in Figs 3D, 3E, 5B and 9C. We have used N = 750 iterations, and the error is of order 10⁻⁶.

4.5 Arnold tongues

The boundaries of the Arnold tongues correspond to the locus of a saddle node bifurcation [62] of the stroboscopic map (9) in the (T, A)-parameter space.

The saddle node bifurcation curves, corresponding to the boundaries of the p : q Arnold tongue, can be numerically computed by identifying the points (θ, T, A) such that the following two conditions hold: (21)

Notice that, although not specified, the stroboscopic map P depends on the parameters T and A. Moreover, since we take modulus T*, we can omit the term pT* in the above equation.

The equations above describe a 1-dimensional curve in , that can be computed using a continuation method. The continuation method is a numerical procedure by which one can find a curve in defined implicitly by a set of (nonlinear) equations G(w) = 0, where is regular enough. Notice that this corresponds to a system with n nonlinear equations with n + 1 variables. The method to find the curve combines a tangent-like approximation as in the Keller’s (pseudo-arclength) method and a modified Newton method (based on a minimization problem with a restriction) to refine it. More precisely, starting from an initial solution w₀ lying on the curve, G(w₀) = 0, the method consists in making a prediction for the next point on the curve by moving along the tangent line to the curve at the point w₀. Then, we correct the approximate point successively by means of a modified Newton method. Since the number of variables is higher than the number of equations, we need to impose the additional condition that the norm of the correction must be a minimum. We solve this problem using Lagrange multipliers. The details of the method can be found in [61, 63].

In our case, the problem consists of three variables (θ and the parameters T and A, after allowing them to evolve according to ) and the two Eq (21). Let w = (θ, T, A) be the unknowns and be the extended flow associated to the phase Eq (8) when T and A are treated as variables, and the associated stroboscopic map. Therefore, we define the function G as (22)

To start the method we must provide an initial seed provided by the rotation number. Notice that the Newton method requires to compute the differential of the map P, which will be computed using variational equations [63]. Moreover, variational equations involve the computation of the derivative of the iPRC, which is only known numerically for a discrete set of points. Taking advantage of the periodic behaviour of the iPRC, we have used the FFT algorithm to compute the derivatives efficiently.

In the case of pulsatile inputs (κ → ∞), the input p(t) becomes a sum of delta functions, i.e. where t_i = t₀ + iT, 0 ≤ t₀ ≤ T. The factor T is included because p(t) was chosen to have average 1 over one period (see Eq (16)), i.e. Thus, the stroboscopic map at time T given in Eq (9) can be written explicitly as: where . Therefore the map has fixed points (modulus T*) as long as θ_n+1 = θ_n + T*, which yields Denoting Z_max and Z_min the maximum and minimum values of the iPRC Z_ei, respectively, we have that the left-hand side and right-hand side borders of the 1:1 Arnold tongue are given, respectively, by the curves Similarly, using that p : q phase-locking corresponds to q-periodic orbits of the stroboscopic map P (equivalently, fixed points of the map P^q) after p turns, we need to impose the condition Thus, the curves corresponding to the left-hand side and right-hand side borders of the p: q Arnold tongue are given, respectively, by the expressions

4.6 Measures of phase-locking and E-cell evoked response

We present four magnitudes that contribute to the description of the phase relationship between the input and the entrained network and describe the changes in firing rate of the E-cells due to the periodic perturbation. These magnitudes are computed for qT-periodic orbits of the perturbed system (1)-(2) with a given function g(t) = Ap(t) of period T, whose existence is guaranteed in the p : q phase-locking regions.

The magnitude Δτ computes the normalized time difference between the maximum of the I-volley r_i and the maximum of the external perturbation p(t), that is, (23) where t_inh and t_p denote the time where the maximum of r_i(t) and the perturbation p(t) are achieved over a cycle of period T, respectively. Notice that if Δτ lies in the interval [0, 0.5), then inhibition follows the input, which may allow an increase of the activity of the E-cells of the target network. In contrast, if Δτ lies in the interval [0.5, 1), then inhibition precedes the input and the latter may be ignored, bringing on a small or negligible effect onto the activity of the E-cells of the target population.

In the 1:2 phase-locking region, where there are two peaks of the perturbation p(t) per one of the inhibitory firing rate r_i(t), Δτ measures the distance between the peak of r_i(t) and the preceding input peak (see, for instance, panel A in S3 Fig). Notice that Δτ lies in the interval [0, 1). If Δτ > 0 but close to 0, the inhibition immediately follows the first peak and precedes by far the second one. If Δτ < 1 but close to 1, inhibition immediately precedes the second peak and follows by far the first one. For intermediate values of Δτ ≈ 0.5, the I-volley is equidistant from both input peaks.

The effects of the perturbation onto the activity of the E-cells of the target population can be measured as well. We define as the ratio between the time-average (over a single period [0, qT)) of the excitatory activity for the perturbed case (A ≠ 0), , and the time-average of the excitatory activity for the unperturbed case (A = 0), , in the following way (24) being (resp. ) the first component of the periodic solution for the perturbed (resp. unperturbed) system.

Notice that if , the external oscillatory input increases the response of the excitatory receiving population. However, the effects of the perturbation onto the receiving population might be also described in terms of enhancement of synchronization in the receiving population. To measure so, we compute the quantities Δα measuring the changes in the maximum and Δσ measuring changes in the half-width.

Following [24], we define Δα as the ratio between the maximum of the excitatory activity r_e(t) for the perturbed case (A ≠ 0), , and the maximum of the excitatory activity r_e(t) for the unperturbed case (A = 0), , that is, (25)

Notice that if Δα > 1, the external oscillatory input increases the maximal response of the excitatory receiving population. Indeed, we may think that changes in the peak height indicate whether the external forcing synchronizes (Δα > 1) or desynchronizes (Δα < 1) the spikes of the target circuit.

The factor Δσ provides the rate change of the half-width of the E-volley due to the external stimulus. It is defined as (26) where and correspond to the two times () at which the firing rate is equal to half of its maximum (). Notice that since we will deal with oscillators of different periods, we must normalize the time difference by its corresponding period.

4.7 Vector strength

When a distractor of different frequency than the primary input is applied, the time-T₁ map of the solution of the phase Eq (8) does not satisfy the conditions to compute the rotation number, since the second input (the distractor) is not T₁-periodic. In this case, we still regard the phase in intervals of T₁ time units (although they are now time-dependent) in order to analyse whether the distractor disturbs the entrainment and by which amount. To do so, we compute the synchronization index (SI) [38], also known as vector strength or Kuramoto order parameter [39]. It is a measure of how clustered are the events over a cycle. To compute the SI one associates to each event a vector on the unit circle with a phase angle and computes the mean vector. The SI is given by the length of the mean vector. That is,

Notice that perfect clustering is obtained when r = 1, whereas if phases are scattered around the circle, then r ≈ 0 (see S6 Fig).

4.8 Computation of the phase shift of the numerical PRC

The Phase Response Curve in Fig 8D provides the phase shift due to the application of a square wave current to the oscillation emerging in the E-I network (1)-(2) when entrained by two identical inputs in anti-phase. It is computed as follows. For every phase t/T₀ ∈ [0, 1) of the oscillator, we apply a square wave current pulse (amplitude 1.5, duration 2 ms) and compute , being T₁ the period of the first cycle after the pulse application and T₀ the period of the entrained oscillator.

The plot also shows, as a function of the phase, if the (entrained) oscillator switches between attended stimulus. To detect whether there is a change in the effective input, we perform the following steps:

1. Before applying the pulse (and once having converged to the periodic orbit) we measure, in a single cycle, the time distance from the maximum of the excitatory firing rate, , to the maximum of each input, and . If or are larger than we consider the preceding input peak. Finally, we take the difference between these two distances, that is, (27)
2. After applying the pulse, we integrate the whole perturbed system for long enough time (we have used 30 periods T₀) so that the transient effects have washed out. Then, we measure again the time distance from the peak of the excitatory firing rate to the peak of each input as in step 1. We take again the difference between these two time distances, keeping the same orientation as in step 1: (28)

A change of sign between d_bf.pl and d_af.pl (i.e. sgn(d_bf.pl ⋅ d_af.pl) = −1) will determine a change in the oscillator’s effective input.

4.9 Numerical methods

Equations for the 8-dimensional system (1)-(2) and for the phase Eq (8) were integrated numerically in Matlab using an explicit Runge-Kutta method of order 4–5 (ode45) with an absolute tolerance ranging beween order 10⁻¹² and 10⁻¹⁶.

Equations of the microscopic model consisting of N_e = N_i = 5000 QIF neurons were integrated using an Euler method with time step Δt = 10⁻⁴. There is a refractory period for each neuron of duration T_ref = 2 ⋅ τ_e,i/V_th, where V_th = 500 is the voltage threshold and τ_e,i = 8 as in [64].

The bifurcation diagram in Fig 2A was computed using the numerical bifurcation analysis toolbox MatCont (a Matlab continuation package) [65, 66].

We used Matlab to analyse and plot the data.

Matlab codes have been released and are available at https://github.com/david-reyner/Matlab-Code.