Fig 1.
Calibration of active properties.
A. Responses of representative neurons to depolarizing current steps from 300 pA to 425 pA in 25 pA increments. A1. mEC PV+ interneuron. A2. Model neuron. Parameters as in Table 1 except EL -80.7 mV, Rinput 83.55 MΩ (gL 11.97 nS), τm 5.45 ms (Cm 65.18 pF), gNA 14426 nS, gKv1 55 nS, gKv3 709 nS, θm -50.7 mV, θh -53.07 mV, θn 8.79 mV, θa = 49.46 mV. B. Population f-I curves. B1. Experimental. B2. Model.
Fig 2.
Calibration of synaptic properties.
Probability densities. A. Chemical Synapses. B. Electrical Synapses.
Table 1.
Parameters for gating variables.
The θ parameters are given for the homogeneous network in Fig 4. These parameters were varied across the network in order to reproduce the variability in f/I curves. The other parameters were held constant for all model neurons.
Fig 3.
Biophysically calibrated levels of heterogeneity are desynchronizing.
Representative spike histograms as a function of time. A1 Hyperpolarizing and B1 Shunting Homogeneous Networks with these parameters: gNA 16805 nS, gKv1 59 nS, gKv3 631.7 nS. EL -72 mV and CM 0.0768 nF. gL was 14.7 nS resulting in an input resistance of 68 MΩ, and gChR was 7 nS, with others as in Table 1. Each neuron received exactly 36 chemical synapses with a strength of 1.65 nS. The synaptic delay was fixed at 0.8 ms. A2 Hyperpolarizing and B2 Shunting Networks with heterogeneity in the active and passive parameters across all 100 neurons. A3 Hyperpolarizing and B3 Shunting Networks with homogeneous neurons but randomly instantiated synaptic connectivity, conduction delays and synaptic conductances. The optogenetic theta drive (bottom) varied sinusoidally at 8 Hz from 0 to 14 nS in all panels.
Fig 4.
Phase Response Curve Explain Synchronizing Tendencies for Hyperpolarizing inhibition.
A. A biexponential inhibitory postsynaptic conductance was used as the perturbation to a single neuron from Fig 3A1 to generate the PRC for hyperpolarizing (red). The strength of an individual conductance was multiplied by 36 to reflect the 36 simultaneous inputs received by a single neuron (left inset) during perfectly synchronous oscillations. The arrows indicate the phase at which an input delayed by 0.8 ms is received in the network. The dashed lines refer to the range of synaptic delays shown in Fig 3A3 and 3B3. The free running period of this neuron is 5.97 ms at a constant ChR conductance of 7 nS, the midpoint of the excitatory theta drive. B. For hyperpolarizing synapses with conduction delays of 0.8 ms, synchrony is stable and attracts from random initial conditions in a single cycle in this raster plot of 20 representative neurons from the 100 neuron network.
Fig 5.
Phase Response Curve Explain Synchronizing Tendencies for Shunting Inhibition.
A. A biexponential inhibitory postsynaptic conductance as the perturbation to a single neuron from Fig 3B1 to generate the PRC for shunting inhibition (green). The dashed green curve shows the normalized change in the cycle after the cycle that contains the perturbation (second order). The strength of an individual conductance was multiplied by 36 to reflect the 36 simultaneous inputs received by a single neuron (left inset) during perfectly synchronous oscillations. The leftmost arrows indicate the phase at which an input delayed by 0.8 ms is received in the network. The dashed lines refer to the range of synaptic delays shown in Fig 3A3 and 3B3. The free running period of this neuron is 5.97 ms at a constant ChR conductance of 7 nS, the midpoint of the excitatory theta drive. B. For shunting synapses, starting from exact synchrony, perturbing even a single neuron (bottom trace) eventually desynchronizes the network. C. If the conduction delay is increased to 1.6 ms in the network with shunting inhibition, synchrony is stabilized and attracts quickly from random initial conditions. B-C are raster plots of 20 representative neurons from the 100 neuron network. Parameters are as in Fig 3 except for ESYN.
Fig 6.
Gap junction connectivity is required for theta nested fast oscillations in heterogeneous networks.
Representative spike histograms as a function of time. A. Networks with hyperpolarizing inhibition. B. Networks with shunting inhibition. A1. and B1. Fully heterogeneous networks (both intrinsic and synaptic heterogeneity) with no gap junctions. A2 and B2. Heterogeneous networks with gap junctions calibrated according to Fig 2B and the text accompanying this figure. A3 and B3. Heterogeneous networks with 2 nS gap junctions with no compensatory reduction in leakage conductance.
Fig 7.
Performance of network with hyperpolarizing synapes is robust to different instantiations of network connectivity.
A. Distinct random network instantiations with constant drive amplitude and frequency. Vertical lines give the standard deviation within a network across theta cycles. A1. Fast frequency with the most wavelet power for each network. A2. Maximum power. B. Same network, variable amplitude 8 Hz drive. B1. Frequency with max power. B2. Maximum power. C. Same network, variable amplitude 4 Hz drive. C1. Frequency with max power. C2. Maximum power. For B and C, vertical lines give the standard deviation for 15 network instantiations across all theta cycles.
Fig 8.
STD induces a preference for theta phases before the peak in networks with hyperpolarizing synapses.
A1. Repeated from Fig 6A2, theta-nested fast oscillations increase on the rising phase of theta stimulation but decrease after the peak due to synaptic depression. A2. Removing short-term depression from the network restores symmetry about the peak for hyperpolarizing networks. B. Circular histogram of onset (blue) and offset (red) phases with and without STD. C. Wavelet analysis. C1. Scalogram of power at each frequency showing how onset and offset phases were determined. C2. Wavelet phase between onset and offset for the bright region of high power bracketed between the blue and red bars in C1. The x and y axis were rescaled to emphasize the region containing theta-nested high frequency oscillations.
Fig 9.
A. Theta drive (bottom trace) synchronizes homogeneous networks provided there is a minimum conduction delay. B. Full heterogeneity disrupts synchrony. C. Adding physiological levels of gap junctions restores synchrony. D. As hyperpolarizing synapses depress during the theta cycle, synchrony is lost. E. Shunting synapses do not synchronize the fully heterogeneous system even at early phases in the presence of synaptic depression and gap junctions.