Multi-Stability and Pattern-Selection in Oscillatory Networks with Fast Inhibition and Electrical Synapses

A model or hybrid network consisting of oscillatory cells interconnected by inhibitory and electrical synapses may express different stable activity patterns without any change of network topology or parameters, and switching between the patterns can be induced by specific transient signals. However, little is known of properties of such signals. In the present study, we employ numerical simulations of neural networks of different size composed of relaxation oscillators, to investigate switching between in-phase (IP) and anti-phase (AP) activity patterns. We show that the time windows of susceptibility to switching between the patterns are similar in 2-, 4- and 6-cell fully-connected networks. Moreover, in a network (N = 4, 6) expressing a given AP pattern, a stimulus with a given profile consisting of depolarizing and hyperpolarizing signals sent to different subpopulations of cells can evoke switching to another AP pattern. Interestingly, the resulting pattern encodes the profile of the switching stimulus. These results can be extended to different network architectures. Indeed, relaxation oscillators are not only models of cellular pacemakers, bursting or spiking, but are also analogous to firing-rate models of neural activity. We show that rules of switching similar to those found for relaxation oscillators apply to oscillating circuits of excitatory cells interconnected by electrical synapses and cross-inhibition. Our results suggest that incoming information, arriving in a proper time window, may be stored in an oscillatory network in the form of a specific spatio-temporal activity pattern which is expressed until new pertinent information arrives

Variety of alternative stable phase-locking in networks of electrically coupled relaxation oscillators.

We studied the dynamics of a large-scale model network comprised of oscillating electrically coupled neurons. Cells are modeled as relaxation oscillators with short duty cycle, so they can be considered either as models of pacemaker cells, spiking cells with fast regenerative and slow recovery variables or firing rate models of excitatory cells with synaptic depression or cellular adaptation. It was already shown that electrically coupled relaxation oscillators exhibit not only synchrony but also anti-phase behavior if electrical coupling is weak. We show that a much wider spectrum of spatiotemporal patterns of activity can emerge in a network of electrically coupled cells as a result of switching from synchrony, produced by short external signals of different spatial profiles. The variety of patterns increases with decreasing rate of neuronal firing (or duty cycle) and with decreasing strength of electrical coupling. We study also the effect of network topology--from all-to-all--to pure ring connectivity, where only the closest neighbors are coupled. We show that the ring topology promotes anti-phase behavior as compared to all-to-all coupling. It also gives rise to a hierarchical organization of activity: during each of the main phases of a given pattern cells fire in a particular sequence determined by the local connectivity. We have analyzed the behavior of the network using geometric phase plane methods and we give heuristic explanations of our findings. Our results show that complex spatiotemporal activity patterns can emerge due to the action of stochastic or sensory stimuli in neural networks without chemical synapses, where each cell is equally coupled to others via gap junctions. This suggests that in developing nervous systems where only electrical coupling is present such a mechanism can lead to the establishment of proto-networks generating premature multiphase oscillations whereas the subsequent emergence of chemical synapses would later stabilize generated patterns

Neuro-economics in chicks: Foraging choices based on amount, delay and cost

Studies on the foraging choices are reviewed, with an emphasis on the neural representations of elementary factors of food (i.e., amount, delay and consumption time) in the avian brain. Domestic chicks serve as an ideal animal model in this respect, as they quickly associate cue colors with subsequently supplied food rewards, and their choices are quantitatively linked with the rewards. When a pair of such color cues was simultaneously presented, the trained chicks reliably made choices according to the profitability of food associated with each color. Two forebrain regions are involved in distinct aspects of choices; i.e., nucleus accumbens–medial striatum (Ac-MSt) and arcopallium intermedium (AI), an association area in the lateral forebrain. Localized lesions of Ac-MSt enhanced delay aversion, and the ablated chicks made impulsive choices of immediate reward more frequently than sham controls. On the other hand, lesions of AI enhanced consumption-time aversion, and the ablated chicks shifted their choices toward easily consumable reward with their impulsiveness unchanged; delay and consumption time are thus doubly dissociated. Furthermore, chicks showed distinct patterns of risk-sensitive choices depending on the factor that varied at trials. Risk aversion occurred when food amount varied, whereas consistent risk sensitivity was not found when the delay varied; amount and delay were not interchangeable. Choices are thus deviated from those predicted as optima. Instead, factors such as amount, delay and consumption time could be separately represented and processed to yield economically sub-optimal choices

Do Altruists Like Equity?

Altruism and inequity aversion are often conceptually interrelated, which implies that altruistic and selfish humans may respond differently to disadvantageous inequity conditions. However, a correlation between altruism and inequity responses has thus far not been directly tested experimentally. We have addressed this question using an experimental paradigm inspired by animal experiments in which adult humans work for real food rewards. We have studied whether subjects' responses to different reward distributions were altered by being exposed to equitable or non-equitable situations. In the control conditions, subjects expressed either a strong altruistic attitude, choosing to work for their partner's welfare in the majority of trials, or mostly rejected this course of action. These purely altruistic and selfish behaviors were also expressed after being exposed to disadvantageous inequity, but priming with equitable conditions significantly reduced their occurrence. This implies an important role of inequity pressure, which is presumably present in modern society, in shaping human-helping attitudes

Phase plane evolution of oscillators involved in the 2-phase behavior in fully connected network.

<p>A. Trajectory of an uncoupled oscillatory cell (black curve) on the phase plane. Evolution along left and right branches of <i>V</i>-nullcline (dashed cubic curve) corresponds to the active and silent phases, respectively. Transitions between the phases occur when the cell jumps up from the left knee or jumps down from the right knee of the <i>V</i>-nullcline (see arrows). The speed of evolution is high during jumps as compared to evolution along <i>V</i>-nullcline when it is inversely proportional to the distance between the actual position of the cell and the <i>W</i>-nullcline (black straight line). Also shown is the <i>V</i>-nullcline for depolarized system (grey cubic curve). B. AP trajectory of 2 cells coupled reciprocally by excitatory synapses. The trajectory (black curve) is identical as in A, except for a right-sided deflection, corresponding to the evolution along the new <i>V</i>-nullcline (grey cubic curve) of the depolarized cell, which is shifted up with respect to the free nullcline (dashed cubic curve). C. AP trajectory of electrically coupled cells. The trajectory of two identical groups of cells expressing anti-phase oscillation (green curve, C1) is compressed comparing to the free cell trajectory (black curve, A) due to continues interaction of two groups all over the cycle. Shown are positions of cells before, during and after the other group's active phase (light grey, black and dark grey circle, respectively) and corresponding nullclines (light gray, black and dark grey cubic curves). Notice shift of the nullclines with respect to the free nullcline (dashed cubic curve) (inserts, C1). Increasing g^el produces gradual compression of the AP trajectory (color curves, C2) and changes a position of cells (black circles) with respect to the knee of <i>V</i>-nullcline (black cubic curve) during a maximal deflection (C3-4, see also C1). When the knee of the nullcline is shifted above a cells' actual position (see black circle and solid cubic curve) a jump up occurs directly from the deflection phase to the active phase (see double arrow) and the AP solution disappears (light blue curve) (C4). Parameters: N = 2 (A-B), 24 (C), N^cc = 23 (C), in B parameters as described in Methods. Phase plane coordinates: <i>V</i> (abscissa), <i>W</i> (ordinate). Trajectories and nullclines were calculated using the software XPPAUT developed by B. Ermentrout (<a href="http://www.pitt.edu/~phase/" target="_blank">http://www.pitt.edu/~phase/</a>).</p

Asymmetrical 2-phase activity pattern.

<p>A. Voltage traces of 2 groups of cells expressing asymmetrical behavior in a fully connected network. Leading is the group of a larger size (22 cells) (black solid curve), the following group of a smaller size (2 cells) (grey solid curve) expresses activity of a diminished amplitude (A). B. Phase plane trajectories of the same groups as in A. C. Phase plane position of deflections of trajectories of both groups in a function of group size ratio in fully coupled (C1) and N^cc2 networks (C2). Parameters: N = 24, g^el = 0.06. Phase plane coordinates: <i>V</i> (abscissa), <i>W</i> (ordinate) (B, C).</p

Effect of network connectivity on the trajectory of cells expressing AP pattern.

<p>A. Deflection of cells trajectory resulting from interaction with the active cells for different connectivity parameter (N^cc). B. Wiring diagram of connectivity for N^cc = 6. Shown are connections close to the border between two groups of cells (vertical line) (see also voltage traces in insert). C. Voltage traces (C1) and phase plane trajectories (C2) of cells belonging to one of the groups. Shown are same cells as in B. D. Consecutive steps of phase plane evolution (D1-D4) of the two groups of cells (solid and dashed lines). Shown are the same cells as in C and analogous cells from the other group. Parameters: N = 24, g^el = 0.22 (A, C-D). Phase plane coordinates: <i>V</i> (abscissa), <i>W</i> (ordinate) (A, C2, D).</p

Multiple activity patterns expressed by network of electrically coupled oscillatory cells.

<p>A. Examples of wiring diagrams of network connectivity for different numbers of closest connected cells N^cc. B. 2-phase and 3-phase patterns expressed by fully connected network. Equally numerous groups of cells (see scheme of cells contribution to different groups, left) express activity in 2 (B1) or 3 (B2) different phases of oscillatory cycle. Note that cells belonging to a given group fire synchronously (see insert, B2). Parameters: N = 24, N^cc = 23, g^el = 0.12 (B).</p

Maximal number of phases in the activity pattern.

<p>A. A general shape of a multiphase pattern trajectory. The trajectory (dark blue curve) is placed between limit values of <i>W</i> (<i>W^up</i> and <i>W^down</i>) (horizontal lines) at which jumps take place. During the silent phase (dotted green curve) a number of deflection may occur (not shown), equal to number of phases in the pattern - 1. The free nullcline is shown as a reference curve (black solid curve). B. Trajectories of cells belonging to different groups expressing 4-phase pattern (B1-2) or 6-phase pattern (B3) during a full active phase of a single cell. For the 4-phase pattern shown are trajectories of cell 1, 7, 13, 19 (black, red, yellow and blue curves, respectively, B1-2). For the 6-phase pattern shown are trajectories of cell 1, 5, 9, 13, 17, 21, (black, dark blue, green, light green and light blue curves, respectively, B3). Notice the effect of shortening the cells' duty cycle (cf. length of trajectories in the silent phase, see color curves in B1 and B2). C. Voltage traces of cells expressing the 6-phase pattern. Only cells connected with a more advanced group are shown. Parameters: N = 24, N^cc = 2, g^el = 0.08. Phase plane coordinates: <i>V</i> (abscissa), <i>W</i> (ordinate). Cells numbers as in Fig. 7B (insert).</p