Oscillations and synchrony in large-scale cortical network models

Abstract

Intrinsic neuronal and circuit properties control the responses of large ensembles of neurons by creating spatiotemporal patterns of activity that are used for sensory processing, memory formation, and other cognitive tasks. The modeling of such systems requires computationally efficient single-neuron models capable of displaying realistic response properties. We developed a set of reduced models based on difference equations (map-based models) to simulate the intrinsic dynamics of biological neurons. These phenomenological models were designed to capture the main response properties of specific types of neurons while ensuring realistic model behavior across a sufficient dynamic range of inputs. This approach allows for fast simulations and efficient parameter space analysis of networks containing hundreds of thousands of neurons of different types using a conventional workstation. Drawing on results obtained using large-scale networks of map-based neurons, we discuss spatiotemporal cortical network dynamics as a function of parameters that affect synaptic interactions and intrinsic states of the neurons.

Fig. 1

Diagram of dynamical regimes produced by map (4). Irregular (chaotic) spiking and spiking–bursting oscillations are typical within the intermediate region outlined by dashed lines and σ_th

Fig. 2

Restructuring of oscillation in map (4) with aα=5.0, bα=4.5, and cα=4.0, as the value of σ increases with the constant slew rate Δσ = 0.00005 per iteration

Fig. 3

Waveforms of x_n for the responses of map (4) to a rectangular pulse of amplitude I_a and duration 1,000 iterations computed for different pulse amplitudes (left) and the input balance parameter β^e (right). The shapes of the pulse are shown below the traces of x_n. The parameter values are α = 4.0, σ = − 0.4, σ^e = 1.0, and a - I_a = 0.630, β^e = 0.224; b - I_a = 0.517, β^e = 0.224; c - I_a = 0.439, β^e = 0.224; d - I_a = 0.394, β^e = 0.224; e - β^e = 0.3, I_a = 0.45; f - β^e = 0.2, I_a = 0.45; g - β^e = 0.1, I_a = 0.45; h - β^e = 0.0, I_a = 0.45

Fig. 4

Phase portraits of the map (4) in the plane of variables u_n = y_n + β_n and x_n explaining the dynamical mechanisms behind the main properties of waveforms for the responses shown in Fig. 3. The samples generated by the map-based model are shown with red dots. To follow the trajectory of the map, these dots are connected with straight yellow lines. a, b, and c Phase portraits for the trajectories shown in Fig. 3h, f, and d, respectively

Fig. 5

The plots of f_s vs n computed for the waveforms shown in Fig. 3a, b, c, and d.

Fig. 6

The dependence of f_S on time n plotted for different values of β^e. The other parameters of the map are the same as in Fig. 3, right panels

Fig. 7

The map responses to a depolarizing (a) and hyperpolarizing (b) rectangular pulse with a duration of 1,000 iterations in the case of a continuously spiking map. The parameters of the maps are α = 4.0, σ = 4.08e − 2, and β^e = 0.3. aI_a = 0.2 and bI_a = − 0.2

Fig. 8

Intrinsic neuronal firing patterns in vivo and in the models. a In vivo data. b Map-based model. Left panels show a fast spiking (FS) neuron, middle panels show a regular spiking (RS) neuron, and right panel show an intrinsically bursting (IB) neuron. The map-based model of the FS cell is given by (9), (10) with α = 3.8, y ^rs = 2.9, β^hp = 0.5, γ^hp = 0.6, g ^hp = 0.1, β^e = 0.1, and amplitude of rectangular pulse A = 1.6E^− 2 (100%). The map-based models of the RS and the IB cell are given by (4), where α = 3.65, σ = 0.06, μ = 0.0005, σ^e = 1.0, β^e = 0.133, and A = 3.0E^− 2 for the RS cell, and α = 4.1, σ = − 0.036, μ = 0.001, σ^e = 1.0, β^e = 0.1, and A = 1.0E^− 2 for the IB cell. The duration of the depolarizing pulse in the map-based simulation is 870 iterations, which is approximately 430 ms

Fig. 9

Waveforms generated by negative rectangular current pulses of amplitude 0.3 and durations of 100, 200, and 400 iterations. The beginning and end of each pulse are shown by arrows. Parameters of the map are μ = 0.002, α = 3.8, σ − 0.15, β^e = 0.6, and σ^e = 1.0.

Fig. 10

Spiking of RS neuron in response to 100 Poisson distributed independent input spike trains with sine-wave modulated mean rate: R = f_c(1 − sin(2 f_mt )), where f_c = 200 Hz, f_m is the frequency of periodic modulation. Sampling rate: one iteration = 0.5 ms. Six different modulation frequencies f_m and 16 trials for each frequency (raster) are shown. Each dot represents a spike. Map-based model of RS cell (left) and experiment with a RS cell of rat prefrontal cortex (right)

Fig. 11

Oscillations in the model of three PY neurons and one IN. Left, excitatory (RS-type) PY neurons are mutually connected with inhibitory (FS-type) IN. Right, increase of excitatory input from PY neurons to IN (from top to bottom) changed the type of oscillations. Different colors indicate three different PY neurons

Fig. 12

Sketch of a two-layer network containing RS PY cells and FS INs. The radius of connection footprint was eight neurons (16 presynaptic neurons) for AMPA-type excitatory PY–IN synapses and two neurons (five presynaptic neurons) for GABA_A-mediated IN–PY synapses

Fig. 13

Resonance phase locking in one-dimensional network model of 256 PY neurons and 64 IN neurons. Frequency corresponding to the highest peak in the power spectrum of the mean field of PY cells’ activity is plotted as a function of synaptic coupling between PY cells and INs. Note several bands corresponding to different resonance-locking modes between FP and PY neurons. The ratio f _PY/f _FP changed from one (lowest band) to four (highest band)

Fig. 14

Spectrograms of oscillation in one-dimensional IN–PY network. Increase of PY–IN coupling was followed by decrease of PY frequency (f _PY) and increase of f _FP and f _IN. Inhibitory INs always fired at the frequency of the FP oscillations. ag_{IN − PY} = 0.0007. bg_{IN − PY} = 0.0015

Fig. 15

Oscillations in two-dimensional network model of 256×256 PY neurons and 128×128 INs. Snapshots of activity in inhibitory, IN, population (top) and cross correlation of local FPs (average activity of 100×100 PY neurons) between two remote spatial areas (bottom) are shown. a Nearly global synchronization–phase locking with zero phase shift, g_{PY − IN}=0.002, g_{IN − PY}=0.0007. b Local synchronization mediated by two rotation spiral waves, g_{PY − IN}=0.0015, g_{IN − PY}=0.0007. c Asynchronous state, g_{PY − IN}=0.002, g_{IN − PY}=0.0015. In top panels, red, depolarizing (spikes), and blue, hyperpolarizing potentials