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 Jab, 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 re and ri 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: Ne = Ni = 5000,
and the rest of the parameters as in (6).
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 re (dashed red) and ri (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).
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 Ve and Vi (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 Ve and Vi, 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.
Fig 4.
Input effects on the E-cell evoked response for a network entrained by coherent inputs.
(A) Evolution of the firing rate variables re (red) and ri (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.
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.
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).
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 re 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.
Fig 8.
Selective communication and switching between attended stimulus.
(A) Time evolution of the excitatory and inhibitory firing rate re (red), ri (blue), respectively for system (1)-(2) receiving two identical inputs of von Mises type in antiphase (κ1,2 = 2 and T = T1,2 = 0.84T*, A1,2 = 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.
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 Ve and Vi (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 Ve and Vi, 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.
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).