This study aims to explore the phase relationship between excitatory and inhibitory neurons in oscillation of neural networks. For the convenience of illustration, we organized neurons into two groups, one for pyramidal cells and the other for interneurons, to highlight how the E-I and I-I loops are formed respectively (Figure 1A), although in practice they may be mixed in the space. To elucidate the mechanism for phase shift clearly, we first considered two extreme scenarios when the network oscillation is controlled by either the E-I loop alone (Figure 1B) or the I-I loop alone (Figure 1C). In the network with the E-I loop alone, pyramidal cells, and interneurons are reciprocally connected, but no connection between neurons of the same type. In the network with the I-I loop alone, interneurons are reciprocally connected among themselves, and they also projected to pyramidal cells but no feedback connection exists. In the network with both the E-I and I-I loops, neurons of the same, or different types are also reciprocally connected with each other. The connectivity is a sparse random graph. Each pair of neurons is connected at probability of 10%. In the simulations for all the above network models, 4000 pyramidal cells, and 1000 interneurons were used.

An interneuron is modeled as single compartment fast-spiking Hodgkin-Huxley type model slightly modified from Wang and Buzsáki (1996), which is written as:

where C_m = 1 nF and I_ext is external input. The leak current I_L = g_L(V−E_L) with g_L = 0.02 μS and E_L = −67 mV. The spike-generation currents are INa=gNam3h(V-ENa), IK=gKn4(V-EK). The variables m, h, and n follow the first-order kinetics, dxdt= ϕx(ax(1-x)-βxx), where subscript x denotes m, h, or n. a_m(V) = −0.1(V + 35) / (−1 + exp(−0.1(V + 35))), β_m(V) = 4 exp(−(V + 60)/18), a_h(V) = 0.07 exp(−(V + 58)/20), β_h(V) = 1 / (1 + exp(−0.1(V + 28))), a_n(V) = −0.01(V + 34) / (−1 + exp(−0.1(V + 34))), β_n(V) = 1 / (1 + exp(−0.1(V + 44)/80)), g_Na = 14 μS, g_K = 1.8 μS, E_Na = 55 mV, E_K = −80 mV, ϕ_m = ϕ_h = ϕ_n = 5.

A pyramidal cell is modeled as two compartments Hodgkin-Huxley type model slightly modified from Geisler et al. (2005), which is written as

where C_m = 0.25 nF. The spike generation currents I_Na and I_K have same dynamics as those of fast spiking interneurons except a_m(V) = −0.1(V + 33) / (−1 + exp(−0.1(V + 33))), β_m(V) = 4 exp(−(V + 58)/12), a_h(V) = 0.07 exp(−0.1(V + 50)), β_h(V) = 1 / (1 + exp(−0.1(V + 20))), a_n(V) = −0.01(V + 33) / (−1 + exp(−0.1(V + 33))), β_n(V) = 0.125 exp(−0.04(V + 44)). The conductances are set as g_Na = 11.25 μS, g_K = 4.5 μS, g_L = 0.025 μS. The reversal potentials are set as E_Na = 55 mV, E_K = −80 mV, E_L = −65 mV. The high threshold calcium current in the dendrite is ICa=gCam∞2(V-ECa), where m is assumed fast variable and is replaced by its steady state m_∞ = 1 / (1 + exp(−(V_d + 20)/9), g_Ca = 0.25 μS, E_Ca = 120 mV. The voltage-dependent, calcium activated potassium current: IAHP=gAHP(V-VK)[Ca2+]/([Ca2+]+30μM). The intracellular calcium follows dynamics: d[Ca2+]/dt=-αICa-[Ca2+]/τCa with α = 4 μM / (ms μA) and τ_Ca = 80 ms, g_AHP = 1.25 μS.

The synaptic currents are mediated byα-amino-3-hydroxy-5-methyl-4-Isoxazolepropionic acid (AMPAR), N-methyl-D-aspartic acid (NMDAR), and γ-Aminobutyric acid (GABAR). Given a spike train, (t_k), in the presynaptic neuron j, the gating variable S, follows the dynamics:

where α denotes G(for GABA), A(for AMPA) and N(for NMDA), respectively. The rising time constants are τ_{G, r} = 0.2 ms, τ_{A, r} = 0.2 ms, τ_{N, r} = 10 ms. The decaying time constants are τ_{G, d} = 5 ms, τ_{A, d} = 2 ms, τ_{N, d} = 100 ms. The postsynaptic neuron i receives synaptic currents I_{i, syn} = I_{i, A} + I_{i, N} + I_{i, G} with Ii,A=(Vi−VE)∑jgji,ASj,A,Ii,N=(Vi−VE)∑jgji,NSj,N1+[Mg2+]exp(−0.062Vi/3.57), and Ii,G=(Vi−VI)∑jgji,GSj,G, where [Mg²⁺] = 1 mM, V_E = 0 mV, and V_I = −70 mV. We only used the NMDA receptors when we simulated the persistent oscillation in Figure 9.

The background noise to a network was modeled as uncorrelated Poisson spike trains delivered to each neuron at a rate of vB = 1kHz (which can be regarded as the net input from thousands of pre-synaptic neurons). The background noise was exclusively mediated by AMPA receptors (AMPARs) with a maximum conductance of 2.48 nS for pyramidal cells and 1.9 nS for interneurons. In the network with both the E-I and I-I loops, the background inputs to pyramidal cell and interneuron are set to be 1.4 and 1 kHz Poisson spike trains, respectively.

The second-order Runge-Kutta method was applied to integrate differential equations with a time-step of dt = 0.02 ms.

The spiking moment of a neuron is taken at the peak of the membrane potential if its value crosses 15 mV, and the size of time bin is dt = 1.0 ms. The instant population firing rate r(t) is given by the number of spikes in the time window [t, t+dt] divided by the number of neurons and dt. The Fourier transformation was applied to the instant population firing rate to obtain the amplitude and frequency of oscillation. By calculating the cross-correlation function between the instant population firing rates of pyramidal cells and interneurons, we obtained the time lag between two discharges, and the phase shift is given by the product of the oscillation frequency and the time lag.

To elucidate the mechanism underlying phase shift clearly, we start to consider a network model whose oscillation is controlled purely by the E-I loop. As shown in Figure 1B, the network consists of two groups, one for pyramidal cells and one for interneurons, and there is no connection between neurons in the same group. Both groups of neurons receive external non-oscillatory excitatory inputs in order to maintain the network activity. It is well-known that such a E-I loop can generate oscillation (Brunel and Wang, 2003; Geisler et al., 2005; Wang, 2010).

We first briefly analyze how the phase shift arises in the E-I loop. In one cycle of oscillation, discharges of pyramidal cells first increase the gating variables of AMPA-mediated synapses (S_A), and subsequently the AMPA-mediated currents depolarize interneurons. Consequently, discharges of interneurons increase the gating variables of GABA-mediated synapses (S_G), inducing GABA-mediated currents, which feedback and finally suppress the activities of pyramidal cells. In such an interaction cycle, the gating variable S_A lags behind the firing rate of pyramidal cells (r_e) by a phase Φ_{P, syn} due to the time consuming of the synaptic dynamics; and similarly the gating variable S_G lags behind the firing rate of interneurons by a phase Φ_{I, syn} (Brunel and Wang, 2003; Geisler et al., 2005). Moreover, the response of a neuron to its input further induces a phase lag, which is denoted as Φ_{P, cell} for pyramidal cells and Φ_{I, cell} for interneurons. The values of Φ_{P, cell} and Φ_{I, cell} depend on the effective membrane time constants of the neurons, and a smaller effective membrane time constant leads to a shorter phase lag (Brunel and Wang, 2003; Geisler et al., 2005). The negative sign of the synaptic currents from interneurons to pyramidal cells contributes a phase lag of 180°. Summarizing all the above phase lags, which forms one cycle, we get Φ_{P, cell} + Φ_{P, syn} + Φ_{I, cell} + Φ_{I, syn} + 180° = 360°. The phase shift from discharges of pyramidal cells (r_e) to that of interneurons (r_i) equals to Φ_{P, syn} + Φ_{I, cell}, which is smaller than 180° according to the above equation, implying that r_e always precedes r_i.

We carried out simulation using the network in Figure 1B to validate the above analysis. As shown in Figure 2A, the raster plots of pyramidal cells and interneurons demonstrate that discharges of pyramidal cells indeed precede that of interneurons. Figure 2B displays that the peaks of S_A, r_i, and S_G fall in the left half side between two peaks of r_e, consistent with the property that the phase shift from r_e to r_i is smaller than 180°.

We further explored how the phase relationship between neurons is affected by the external excitatory inputs to the network and the synaptic conductance between pyramidal cells and interneurons. We found that: (1) increasing the excitatory inputs to interneurons (I_{ext, i}) alone reduced the phase shift between pyramidal cells and interneurons (see Figure 3A1). This is due to that a stronger excitatory input to an interneuron leads to an effectively shorter membrane time constant, which speeds up the response of interneuron to its input; (2) increasing the excitatory inputs to pyramidal cells (I_{ext, e}) alone enlarged the phase shift (see Figure 3A2). This is due to that a stronger excitatory input to a pyramidal cell leads to an effectively shorter membrane time constant, which makes the pyramidal cell discharge earlier in one cycle; (3) varying the synaptic conductance from interneurons to pyramidal cells has little influence on the phase shift (see Figure 3B1); (4) increasing the synaptic conductance from pyramidal cells to interneurons decreases the membrane time constants of interneurons effectively, which leads to a reduced phase shift (see Figure 3B2).

We further considered a network model whose oscillation is purely controlled by the I-I loop. As shown in Figure 1C, interneurons are reciprocally connected, and they project to pyramidal cells without feedback. The previous study has shown that characteristic oscillation can arise from the I-I loop via mutual inhibition between interneurons, if the external excitatory input is sufficiently strong (Wang and Buzsáki, 1996). In such a case, the excitatory input depolarizes interneurons to fire, which release neurotransmitters GABA to activate GABA-receptors at the postsynaptic neurons, suppressing the activities of interneurons. Those interneurons will not discharge until the external input activate them again. This process is repeated and the network oscillates.

We can decompose one circle of oscillation into different components. As shown in Figure 4A, the gating variable of synapse conductance S_G follows the interneurons' firing rate r_i with a phase lag Φ_{I, syn} due to the synaptic dynamics (Brunel and Wang, 2003; Geisler et al., 2005). The response of interneurons to their inputs induces another phase lag Φ_{I, cell} (Figure 4C). Consider that the negative sign of inhibitory currents contribute to a phase lag of 180°, we have Φ_{I, cell} + Φ_{I, syn} + 180° = 360°. Thus, quicker neuronal response leads to a smaller Φ_{I, cell} and a larger Φ_{I, syn}; whereas, slower neuronal response leads to a larger Φ_{I, cell} and a smaller Φ_{I, syn}. Particularly, the stronger external excitatory input leads to a shorter effective membrane time constant, which means a smaller Φ_{I, cell} and then a larger Φ_{I, syn} (Figure 4B1); whereas, stronger mutual inhibition leads to a longer effective membrane time constant, which implies a larger Φ_{I, cell} and a smaller Φ_{I, syn} (Figure 4B2).

In a network whose oscillation is controlled by the I-I loop, the oscillatory responses of pyramidal neurons are driven by interneurons (see Figure 1C). We can evaluate the phase shift between pyramidal cells and interneurons accordingly. The gating variable S_G induces a phase lag Φ_{I, syn} with respect to the firing rate r_i. The response latency of pyramidal cells to inputs induces a phase lag Φ_{P, cell}. Consider that the negative sign of inhibitory currents induces a phase lag of 180°, the total phase shift from r_i to r_e = 180° + Φ_{I, syn} + Φ_{p, cell} (Figure 5D). Therefore, we have: (1) if pyramidal cells have small effective membrane time constants and short response latency, such that the total phase shift is larger than 180° but smaller than 360°, pyramidal cells precede interneurons in oscillation (Figures 5A,D1; note that here we define “leading vs. lagging” based on the amount of phase shift, rather than the actual causal relationship between neuronal responses); (2) if pyramidal cells have large effective membrane time constants and long response latency, such that the total phase shift is larger than 360°, pyramidal cells follow interneurons in oscillation (Figures 5B,D2). Figure 5C2 presents an example that by increasing the external input to pyramidal cells (i.e., larger synaptic conductance G_{ext, e}), which effectively reduces the membrane time constant and the response latency (Geisler et al., 2005), the phase relationship between r_e and r_i can change from lagging to leading. Figure 5C1 presents an opposite example that by increasing the inhibition strength from interneurons to pyramidal cells (i.e., larger G_ie), which effectively enlarges the membrane time constant, the phase relationship between r_e and r_i can change from leading to lagging. Overall, depending on the parameters, the phase of pyramidal cells can either precede or lag behind that of interneurons in a network oscillation determined by the I-I loop.

In general cases, excitatory and inhibitory neurons in a network are normally reciprocally connected, forming both E-I and I-I loops (Figure 1A), and the network oscillation is a result of the interplay between two loops. Depending on the parameters, the network oscillation may be dominated by either the E-I loop, or the I-I loop, or a mixture of both, and display different phase shift characteristics. We explored how the magnitude of external excitatory inputs (representing modulations from other brain areas) and the strength between excitatory and inhibitory interactions affect the phase relationship between neurons.

We first explored how the external excitatory inputs to interneurons affect the phase relationship between neurons. For comparison, we set a baseline oscillation in which the phase of pyramidal cells leads that of interneurons, as shown in Figure 6. The network parameters fall in the regime where the responses of interneurons are mainly driven by pyramidal cells, and the E-I loop dominates the network oscillation. We then increased the external excitatory inputs to interneurons gradually (I_{ext, i} from 3.5 to 17 kHz) while kept other parameters invariant, and observed a transition in the network oscillation from E-I dominating to I-I dominating. Figure 7A presents the results, which are: (1) with the increase of the external inputs to interneurons, the phase shift of pyramidal cells with respect to interneurons transits from leading to lagging (from about 80° to about −60°, Figure 7A1). This is because when the excitatory inputs to interneurons become sufficiently strong, the I-I loop dominates the network oscillation. (2) The frequency of the network oscillation increases from about 150 Hz to about 300 Hz with a jump near the transition point between two operating regimes (Figure 7A2). (3) Pyramidal cells and interneurons oscillate at the same frequency when the network oscillation is at a single mode, however, they oscillate at different frequencies near the transition point (Figure 7A2). (4) The oscillation amplitude of pyramidal cells decreases from 50 Hz to few hertz with the increase of the external inputs I_{ext, i} (Figure 7A3), but the oscillation amplitude of interneurons has little change.

Similarly, we explored how the external excitatory inputs to pyramidal cells affect the phase relationship between neurons by keeping other parameters unchanged. As shown in Figure 7B, starting from a regime where the I-I loop dominates, we increased the external inputs I_ext,e gradually (from 4 to 20 kHz), and observed a transition from I-I loop dominating to E-I loop dominating in the network oscillation. The results are: (1) with the increase of the external inputs, the phase shift of pyramidal cells with respect to interneurons changes from lagging to leading (from about −70° to about 80°, see Figure 7B1). (2) The frequency of the network oscillation increases when either the I-I loop or the E-I loop is dominating, but there is a big drop at the transition point between two oscillating modes (Figure 7B2). Particularly, near the transition point, two or more cycles of interneurons oscillation emerge from the original one cycles of oscillation, while pyramidal cells keep the original cycle. This leads to different oscillation frequency of pyramidal cells and interneurons near the transition point. (3) The oscillation amplitude of pyramidal cells increases from a few Hz in the I-I loop dominating mode to about 80 Hz in the E-I loop dominating mode (Figure 7B3).

We further explored how the connection strengths between excitatory and inhibitory neurons affect their phase relationship in oscillation. We first increased the inhibition strength from interneurons to pyramidal cells (by increasing the synaptic conductance G_ie), and found that the network oscillation can change from E-I loop dominating to I-I loop dominating. Figure 8A presents the results, which are: (1) with the increase of G_ie, the oscillating phase of pyramidal cells with respect to interneurons changes from leading to lagging (from 120° to −60°, Figure 8A2); (2) with the increase of G_ie, the frequency of network oscillation first decreases gradually from 260 to 120 Hz, and then experiences a sharp upsurge near the transition point when the network oscillation changes from E-I loop to I-I loop dominating (Figure 8A2); (3) The average firing rate of pyramidal cells decreases with the increase of G_ie, whereas, the oscillation amplitude of pyramidal cells increases first and decreases subsequently (Figure 8A3).

Similarly, we explored how the connection strength from pyramidal cells to interneurons affects the phase relationship. Increasing the synaptic conductance G_ei has 2-fold effects: on one hand, it increases the excitation to interneurons from pyramidal cells, but on the other hand, it also increases the feedback inhibition to pyramidal cells from more active interneurons. These two effects largely cancel each other. Therefore, we observed that: (1) with the increase of G_ei, the phase shift from pyramidal cells to interneurons decreases from about 30° to slightly larger than zero (Figure 8B1). (2) The frequency of network oscillation decreases from about 200 Hz to about 150 Hz (Figure 8B2). (3) The oscillation amplitude of pyramidal cells has only a minor change (Figure 8B3).

In the previous sections, we have considered that the neural networks receive external excitatory inputs and the latter contributes to the network oscillation. It is known that a neural network can also retain oscillation without relying on external drive, but rather depending on the positive feedback interaction between pyramidal neurons via slow NMDA receptors (Tegnér et al., 2002). This persistent activity has been observed in several cortical areas and is proposed to play important roles in higher cognitive functions such as working memory (Fuster and Alexander, 1971; Funahashi et al., 1989) and attention (Ibos et al., 2013). Obviously, in persistent oscillation, the E-I loop always dominates, which drives the I-I loop if the latter exists, and consequently, the phase of pyramidal cells always precedes that of interneurons. We simulated a network consisting of 800 pyramidal cells and 200 interneurons, and included NMDA receptors between pyramidal cells. The results are shown in Figure 9, which are: (1) the network oscillation persists after even the external input is removed; (2) the variation of the synaptic conductance from interneurons to pyramidal cells has little effect on the network oscillation (Figure 9B); (3) the increase of the synaptic conductance from pyramidal cells to interneurons makes interneurons discharge earlier in one cycle (Figure 9C1) and has minor effects on the frequency and amplitude of the oscillation (Figures 9,C2,C3).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer YQ and handling Editor declared their shared affiliation, and the handling Editor states that the process nevertheless met the standards of a fair and objective review.