Fig 1.
Typical dynamics of the model neuron, Eq (1), with and without the graded-persistent activity (GPA) when successive pulse external stimulus (top panels) are given.
For a normal neuron (left panels, ), the cell activity (left middle panel) decays rapidly when the pulse input is terminated as its auxiliary variable (left bottom panel) is kept small. By contrast, for a GPA neuron (right panels,
), the cell activity (right middle panel) is kept almost constant, even during intervals between pulse external inputs. The sustained cell activity gradually increases in response to repeated depolarizing pulse inputs and decreases in response to subsequent hyperpolarizing pulse inputs. These sustained activities are supported by the slow decay of the auxiliary variable (right bottom panel).
Fig 2.
Typical dynamics of neurons in the random network, Eq (2), for values of the coupling strength g and the ratio of the GPA neurons p.
Panels are arranged such that the coupling strength g increases from left to right (), and the ratio of the GPA neurons increases from top to bottom (
). For the larger value of the ratio of GPA, p, the transition from the silent to the chaotic state occurs at the smaller value of g. Other parameters are N = 3000,
,
, and
.
Fig 3.
Behavior of graded persistent neurons and other normal neurons in a network, Eq (2), operating in the chaotic regime.
(a) Temporal profiles of the activity of neurons in each population. (b) Autocorrelation functions of the time series for individual neurons. (c) Autocorrelation functions averaged over neurons in each population. (d) Density distributions of the correlation times of neurons in each population, obtained directly from the autocorrelation functions in (b). Parameters N = 5000 p = 0.5 and g = 2.0 are used. All results indicate that graded persistent neurons and other normal neurons behave similarly in the network under chaotic dynamics.
Fig 4.
Maximum power spectrum of the network dynamics for values of the ratio of the GPA neuron p (horizontal axis of each panel) and coupling strength g (vertical axis of each panel).
Red curves are the theoretical prediction of the transition point, gc, given by Eq (15). a: and
are fixed and
increases from left to right (
). b:
and
are fixed and
increases from left to right (
). c:
and
are fixed and β increases from left to right (
). Numerical simulations are performed with N = 3000 for all panels.
Fig 5.
Eigen spectrum in the complex plane for the Jacobian at the trivial fixed point of the network dynamics.
Dots in each panel indicate the eigenvalue of the linear stability matrix on the complex plane. Blue and red dots indicate eigenvalues with negative and positive real parts, respectively. Panels are arranged such that the GPA neuron ratio, p, increases from left to right, and coupling strength, g, increases from top to bottom. The thick vertical line in each panel is the imaginary axis. Panels shaded by red background mean the coupling strength of the network is above the transition point predicted theoretically, i.e., g > gc. Other model parameters are and
.
Fig 6.
The maximum Lyapunov exponent of the network dynamics.
a: The numerically estimated maximum Lyapunov exponent, averaged over ten realizations of random initial conditions, for the model network as a function of the coupling strength g. Line colors indicate the value of p (from the far-right purple curve for p = 0.0 to the far-left yellow curve for p = 1.0). Dashed vertical lines indicate gc. The number of neurons in the network is fixed at N = 3000. b: The same Lyapunov exponent for different values of N with p = 0.8. Blue, orange, green, and red lines correspond to networks of N = 100, 500, 1000, and 3000, respectively. Error bars show standard deviation.
Fig 7.
Maximum power spectrum of the network dynamics with Gaussian heterogeneous adaptation for values of model parameters.
The horizontal and vertical axes are and g, respectively. The red curves show theoretical prediction gc obtained using the method developed in this study The white dashed lines indicate
that are derived from a naïve approximation of the heterogeneity.