Robustness of Persistent Firing in a minimal recurrent network of Working memory

Figure 13. Phase trajectories of a HH neuron clos to saddle node bifurcation in presence of high (a and b) and low (c and d) noise. Blue lines are the subthreshold activities. Dashed and solid black lines are the unstable and stable limit cycles respectively. b (c) is the magnified version of a(c).

3.2.3 Future work

The idea of ISR phenomenon is still young and needs to be more studied theoretically and experimentally. The systems which had been considered till now are kind of oscillatory circuits that unlike systems under discussion of SR are not supposed to detect any external signal. They produce self-sustained activity and the notion of SNR doesn't really make sense for them. Their performance depends on what mode of activity is favorable in the bi-stable system. For example, if termination of periodic activity in presence of small perturbations is desired, then the ISR effect might be considered advantageous. Alternatively, if it is required to extinguish the oscillatory activity of a system at a particular noise intensity ISR phenomenon can perfectly implement such filtering. On the other hand, as long as robustness of oscillations in presence of noise is demanded, ISR effect will be troublesome, of course.

In this thesis we will show that our intended minimal model of WM has the required dynamical structure for occurrence of ISR. As mentioned in chapter 2, the stored information in WM, must be cleared steadily within a short period after accomplishment of the task. Therefore, it is proposed that randomness in neural circuits of WM is advantageous because it can help the system to frequently wipe out the memory.

Considering a range of parameter variations, the robustness of sustained activity will be investigated in the circuit of two QIF neurons. Occurrence of ISR effect will be verified for networks exposed to Gaussian white noise as well as coloured noise. We haven't observed stronger ISR effect by coloured noise as it had been reported by Guo (2011) in a HH model, but it will be shown that optimal values of noise grow relative to the level of autocorrelation of the coloured noise.

Probability of stopping the activity by noise, Firing Rate Histogram, Inter Spike Interval Histogram, and Concavity Width of the ISR curves, are the ISR properties that will be studied in this work.

Chapter 4. Simulations and results

Two QIF neuronal models with the following system of equations (3) are employed to form a minimal network of synaptically connected neurons. Each QIF neuron is individually a saddle node type-1 oscillator with the after spike resetting mechanism. Increasing the input currents while they are coupled, moves the circuit through a global bifurcation and leads to emergence of an antiphase bistable system.

(3)

X and Y are the membrane voltages of each neuron, and as soon as their values exceed V_peak they will abruptly set back to a reset point, V_reset which is chosen to be always smaller than the rest state of the neuron. This leads to an undershoot in the evolution of the action potential following an exponential increase toward rest. I_ext is always a constant negative input employed for keeping the cells excitable in absence of synaptic connection. Therefore, as long as the synaptic input current, I_syn, is zero, the roots of each neuron's equation would stay on the unstable and stable equilibrium points, and respectively.

In figure 14 the function of a typical single QIF model is illustrated in panel a. In panel b simulation of the membrane voltage alterations are plotted for four different initial states of a single QIF neuron. It is seen that individual uncoupled neurons are excitable. They produce single action potentials only when they are triggered externally. Selected parameters for this simulation are given in table 1.

Figure 14. a) Function of the derivative equation of a single QIF neuron, b) evolution of membrane voltage of a single QIF with different initial states.

Table 1 Selected parameters for simulation of QIF model

Next, QIF neurons are coupled by a time dependent synapse with the following equations, (4).

() is the time point at which neuron () hitts the peak and , is the synaptic time constant. The function is a standard step function which allows the synaptic current to initiation at time . Consequently, as soon as X (Y) produces an action potential, it provides a synaptic current as an input to Y (X).

J is the synaptic weight connecting the neurons. If the synaptic force after a spike by one neuron is sufficiently large to convey the other neuron across its threshold, this procedure will continue and neurons will reciprocally excite each other leading to formation of a limit cycle on the phase plane. This limit cycle will emerge at a global bifurcation point that is marked `J*' in this thesis. Consequently, this network which had a single attractor at `-a', before, will bifurcate at J* and launch into a new bi-stable mode, possessing the former stable rest, plus a Stable Limit Cycle (SLC). As a result, the two neurons, that were individually excitable, will be able to produce self-sustained regular activity when they are coupled, provided that an external stimulus triggers the first action potential in one of them.

Since, in the conditions of sufficient synaptic connection, these neurons can continue firing after disappearance of the triggering stimulus, behavior of the model is somehow analogous to the persistent activity of prefrontal neurons during the delay phase of a working memory task.

4.1.1 Theta Neuron

For investigating the phase space of this system, the dynamical evolution of voltage values should be converted into dynamics of a phase variable. Here, the и-neuron equations are employed for this purpose. -neuron, is a phase model featuring the dynamics of any type-1 neuron that is able to produce a wide range of frequencies. A neuron is described by a phase variable which indicates the location of voltage along the trajectory of action potentials. By a simple substitution , QIF will be represented as a neuron. `2.atan(x)' and `2.atan(y)' are the phase variables that always lie in an interval between - and . The ()-plane will be exploited for illustrating the phase portrait of the coupled system. Equation (5) indicates a single standard Theta model. For more details about this canonical model look at (Gutkin, & Ermentrout, 1998).

4.1.2 Phase Space

The circuit of two QIF neurons have four fixed points; S_xx: (-a -a), S_xy: (-a a), S_yx: (a -a), and S_yy: (a a), which are depicted with green circles in figure 15. This plane is actually a periodic torus and orbits revolve it from left to right and from bottom to up. With the same parameters as in table 1, trajectories of the phase variables are plotted starting from four different initial states. As it is observed, in absence of synaptic connection ( ), all trajectories have terminated at the stable attractor S_xx.

Figure 15. Dynamics of the uncoupled network. a) Phase trajectories for 5 different initial states. b) corresponding membrane voltages.

S_xy and S_yx are the saddle nodes and S_yy is the unstable or repeller node of this dynamical system. Synaptic currents provided by a synaptic connection induce an additive force to the network's variables. As a result, by a small positive J straight paths of phase trajectories to stable `S_xx', divert toward the saddles. This will slightly enhance the membrane voltage under the threshold, but, as it is shown in figure 16, this growth doesn't help reaching the saddle. Thus, trajectories have been eventually attracted by the stable point. The selected parameters for simulations shown in this figure are similar to the previous one, except for the synaptic weight, that is chosen to be J = 5.

Figure 16. Dynamics of the weekly coupled network, with J = 5. a) Phase trajectories for 5 different initial states. b) corresponding membrane voltages.

4.2 Switching On the Persistent Activity

In the conditions discussed above neurons were either uncoupled or just weekly coupled. The system was mono-stable with constituent neurons that were in their excitable mode. In figure 17, it is indicated that how gradual increment of J varies the shape of orbits on the plane, so that it ends up to formation of a limit cycle. In this figure synaptic strengths have increased from J = 0 to J = 8 with a unit step size from panel a to i. Thus, panel a corresponds to the phase portrait of the uncoupled network. Panel b to f indicate the phase portraits for subcritical values of J. At J = 6 in panel g, the first spike produced by neuron Y has become able to create a sufficiently large synaptic current for triggering neuron X. And neuron X has also triggered neuron Y in turn. This reciprocal excitation has shaped a limit cycle on the plane. Therefore, the critical J* for this global bifurcation, where another attractor -the limit cycle- has emerged, must be somewhere between J =5 and J = 6. The exact value of J* will be computed in later sections. Here the important issue is bi-stability of this network due to coexistence of the stable node and the SLC. These attractors, respectively, simulate the silent and persistent activity modes of PFC neurons in WM.

The new-born limit cycle at J* is too fragile and very sensitive to perturbations. Greater synaptic weights, J > J*, as shown in panels h and i keep the limit cycle farther from the saddle nodes. Since saddle nodes are directors of the structure of attraction basins, it is expected that these limit cycles are more robust and have larger attraction domains.

But what is important to notice is that the concept of attraction domain in non-autonomous systems is not analogous to autonomous systems. In the first case whether a typical point on the phase plane is in the attraction domain of one attractor or not is a static property, while for the latter system this property changes with time. In simple words, if a phase variable standing on a determined location of the space will be attracted by the stable rest point or by the limit cycle varies time to time. In this context, talking about growth of attraction does not only involve spatial structure of the phase space but also involve time lapse. In general, when we say that the limit cycle's attraction domain is expanding, it means that for a full period the amount of times that a phase variable will be attracted by the limit cycle is greater than before.

Figure 17. Formation of the limit cycle on the phase plane from subcritical to supercritical synaptic strengths. green nodes are the fixed points and black orbits are the phase trajectories of the system. a) J = 0 (uncoupled), b) J = 1, c) J = 2, d) J = 3, e) J = 4, f) J = 5, g) J = 6, h) J = 7, i) J = 8

For the simulations in figure 17, initially neuron X is assigned to the reset point (X(0) = -8) and neuron Y is slightly above its threshold (Y(0) = 1.01). This initial condition is chosen to provide the initiation of the first spike and the first synaptic current to turn on persistent firing. However, it must be noted that the required synaptic strength for bifurcation highly depends on the initial state of the system.

In figure 18, the critical J* required for switch on is color mapped for initial states, i.e. X(0) and Y(0), varying in the interval [-10 80]. Simulations has run for J increasing from zero to 16 with a unit step size. The yellow color in the square region on the lowest left part of the figure signifies that switching on the activity is impossible, because, trivially, in this region both X and Y are under threshold and none of them is disposed to initiate any spike. The smallest required J is about 6, on the lines where one neuron is above threshold and the other is slightly below threshold. The more synchronous are X and Y, the stronger J is required to turn on the activity. This fact is apprehensible from the expansion of more greenish colors close to the diameter of the plane. Such phenomenon is somehow related to the fact that in a network with recurrent excitation, firing is necessarily asynchronous which was proved theoretically by Gutkin et al (2001).

Figure 18. x and y axes indicate the possible initial states of membrane voltage for neurons, i.e. X(0) and Y(0). Color values illustrate the minimum required J for bifurcation relative to different initial state.

Here also, as expected, X and Y activate asynchronously. Phase portrait of the system in activity mode is illustrated in figure 19.a and corresponding spike trains and synaptic currents are plotted in panel b. All parameters and initial conditions are the same as parameters given in table 1, and J = 6.

Figure 19. Dynamics of the coupled network with synaptic strength, J = 6. a) Phase portrait at bistable mode. b) The first and third traces are spike trains. The second and last traces are corresponding synaptic currents.

4.2.1 External current effect

The critical synaptic strength for transition from silent to activity mode, depends also on the constant negative drive I_ext. The more negative is the external current, the farther is each neuron from its saddle node bifurcation, and thus farther from the oscillatory mode. It raises the expectation that for more negative values of I_ext, a larger synaptic connection will be needed for occurrence of bifurcation. This phenomenon might be easier to grasp if we consider that for larger negative values of I_ext the distance between `a' and `-a' will be greater, So, the synaptic current should push the orbits harder for crossing this distance.

To confirm this effect, the same procedure as done for figure 18 has been repeated with four different values of I_ext and results are shown in figure 20. I_ext is decreasing from -1 to -4 from panel a to d respectively. To make the plots comparable with each other, all values of J are scaled into an interval between zero and 100. Yet the general structures of the maps are similar, more yellowish colors have appeared gradually from panel a to d, signifying the larger values of J required for activity initiation in the condition of more negative external drive.

Figure 20. Color maps of minimum required synaptic strengths for bifurcation relative to the initial state, for four different external currents. a) I_ext = -1, b) I_ext = -2, c) I_ext = -3, d) I_ext = -4

4.3 Switching Off the Persistent Activity with excitation

Relying on available literature that were reviewed in chapter 2, it is predicted that inhibition can also stop the persistent activity in the degree that it synchronizes the network's elements, as excitation did. To check this prediction, an inhibitory pulse is applied to both neurons in the active network at different time points, and firstly the minimum inhibition (i.es the maximum negative value) required for transition to the silent mode is computed.

The same as before, for J = 6 that is very close to bifurcatio, the system is highly sensitive and its activity soon ceases with tiny inhibitions. Also, the stronger is the synaptic connection, the larger inhibition is required for transition to silent mode. However, unlike the previous case, even networks with very big J values are finally stoppable with brief inhibitory pulses. The color mapped results are shown in figure 25 for J=6 to J=17. Only in the last four plots, the dark blue areas of the map show the trials in which activity has never stopped by inhibitions up to -100.

Figure 25. Color maps illustrate the minimum inhibitory kick size for switching the network from activity to silent mode. Lower traces indicate the corresponding phase state of the system at arrival time of the transient inhibition. For all networks I_ext = -1, a) J = 6, b) J = 7, c) J = 8, d) J = 9, e) J = 10, f) J = 11, g) J = 12, h) J = 13, i) J = 14, j) J = 15, k) J=16, l) J = 17

One more fact that can be grasped implicitly by comparing these results is that the structure of attraction basins are not generic for networks different parameters. If we track the blue areas of the colormaps that correspond to the least robust regions od the SLC, it is seen that their locations reletive to the phase are changing for each case.

In the next step, binary modes of the network (blue: Stop, and Yellow: remain active) for a full range of amplitudes of inhibitory pulses are plotted in figure 26. There are still some noncontinuities seen in the plots, but in general, these results sugest that switch-off is more trustable via inhibitory pulses than excitatory ones. Because the blue ares are more bulky in comparison to the similar networks in fugure 24, and also they even for strongly coupled networks with large j values.

Figure 26. Color maps illustrate the state of the network after an inhibitory kick. The y-axes of color maps show the kick size and the x-axes show the index of time points at which the transient inhibition arrives. Lower traces indicate the corresponding phase state of the system at arrival time of the transient inhibition. For all networks I_ext = -1, a) J = 6, b) J = 8, c) J = 10, d) J = 12, e) J = 14, f) J = 16

In general, inhibitory transient stimuli are appropriate tools for clearing a memory that is sustained in a recurrent network of excitatory coupled cells. This mechanism is highly trustable for networks near bifurcation. When synaptic weights go far beyond J*, the limit cycle become more robust and it will be more difficult to terminate the persistent firing. However, synchronizing neurons through inhibition seems more trustable than excitation for switching from the activity to the silent mode.

4.5 Firing Rate and Type of Activity

As discussed earlier, QIF model is a type 1 neuron which its firing rate smoothly raises from zero as the input current increase (Hodgkin, 1948; Gutkin et al., 2008a). It is trivial that for a single QIF neuron there exist a saddle node bifurcation point at I = 0 that switches the neuron from excitable to oscillatory mode. Here, the input current to the constituent neurons of our system is composed of two terms. First, I_ext, that is a constant negative drive equal to both neurons. And next, the synaptic current, I_syn, provided by J, that has a periodic nature and varies with time. I_syn is always positive and as long as the activity is not too fast, it has a zero minimum and a maximum at J.

Regarding the behavior of coupled neurons, first of all, we can clearly predict that while I_ext is positive, the network's behavior is necessarily periodic. Besides, due to the intrinsic properties of type 1 cells, it is expected that the bigger is each of the input current terms, J or I_ext, the higher frequency of firing will the network exhibit in the active mode. In figure 27.a Fr-J function is plotted for ten different constant external currents. This figure shows that when I_ext is close to the intrinsic bifurcation of the individual neurons, zero, a weaker synapse can push the network from silent into the active mode.

In this figure, unlike the Fr-I curves of the single neurons, the jump from zero to a positive Fr, is not due to saddle node bifurcation. In fact, J is just lifting the quadratic function of each neuron up and down periodically. Equilibriums still exist there at , but they are just shifting back and forth due to the force of I_syn. So, the coupled excitable neurons reciprocally trigger each other and form a regular pattern of firing that corresponds to a stable limit cycle in the phase space.

In addition, it is shown in this figure that the more negative is I_ext, the more type 2 behavior has the Fr-J curve, while the less negative it is, Fr raises more smoothly and the curve looks more type 1.

Figure 27. a) Fr-J curve of the coupled network for ten different fixed I_ext, b) Firing rate in the parameter space of I_ext-J.

In figure 27.b, a full range of firing rates for different pairs of J-I_ext are shown in a color map plot. The dark blue region in the lower left part of the plane indicates the parameter space where the system is mono-stable and cells are excitable. Firing rate is zero in this area and the boarder of bifurcation for switching on the persistent activity is clearly observable. As expected after the bifurcation, firing rate is growing on both axes.

If the speed of activity goes up as a result of increasing either J or I_ext, at some point the synaptic current will not decay suf?ciently fast compared to the interval between adjacent spikes of the neurons. Consequently, a DC term (I_DC) will appear in the synaptic current signals. We believe that something similar to a saddle node bifurcation for the coupled network must occur where the total constant input current to both neurons equals to zero. That is where the DC offset of the synaptic current grows so far as it completely cancels out the negativity of the external current. In such critical point the four equilibriums of the system converge, collide, and annihilate each other. Therefore, this system is able to show an extremely robust periodic behavior even when its constituent neurons are at their excitable mode.

To illustrate the effect of increasing J on the pattern of synaptic currents, these signals are plotted for four different networks with fixed I_ext = -1 in figure 28. The speed of activity is too low in panel a where J = 6 and synaptic currents have no overlap with each other. As J has increased to 8 in panel b firing rate has also increased, but the peak of I_syn is still equal to J and its minimum is zero. In the lower traces c, and d the speed of activity is too high and DC offsets are clearly observable in synaptic signals. In these cases, after a few spikes, peaks of I_syn have exceeded J and reached J+I_DC.

Figure 28. Synaptic currents for 4 different networks. Firing rate increases due to increment of J from panel a to d. When the speed of activity is too high, synaptic time constant is relatively slow and a DC offset emerges in the synaptic current signals. I_ext is fixed at -1 for all the networks. a) J = 6, b) J = 8, c) J = 12, d) J = 16

4.5.1 Activity Regions

The first requisite for modelling mechanisms of WM in the brain, is a bistable system apt to switch between oscillatory and rest states. We believe that dynamical properties of a system which has reached a stable activity with I_DC I_ext changes in a way that transition from activity to silent mode become somehow impossible. In such condition, since the effect of negative drive is absent, equilibriums will gradually vanish after switch on. Therefore, after a transient period all the phase plane will be covered by the attraction domain of the SLC. The only way to switch off the activity in such condition is applying sufficient inhibition that can decrease all the synaptic currents and bring them back to zero level.

To achieve the exact range of parameters that lead to a reasonable bi-stability in the system under our discussion, first, we have assumed that the synaptic time constant is always fixed at = 5 ms. Second, we only consider the negative external currents to make sure that the single cells are not oscillatory and the system possesses four equilibriums, one of which is stable. Then, J is treated as a bifurcation parameter and for I_ext ranging from -10 to -0.1 two critical points are calculated for the system.

1- A critical synaptic weight, J*, that turns the activity on and cause the system to bifurcate from mono-stability to bi-stability. In fact, for J < J* the system has only a stable node that is S_xx, equivalent to rest state. At J* a SLC emerges and coexist with the former stable rest point.

2- Another critical synaptic weight, J**, at which the absolute value of I_ext equals to the DC offset of synaptic current and leads to a saddle node bifurcation. For J > J**, provided the system is already triggered, and the transient period has passed, the equilibriums vanish and a single attractor remains in the phase space, that is the limit cycle. In this condition, all the phase plane is the attraction domain of the SLC. However, as long as there is no stimulus to turn on the activity, equilibriums are determined by I_ext and the system is silent. Therefore, after this critical point, the system is still considered bi-stable but in a short time after it is switched on, kind of a local bifurcation occurs, rest points vanish, and the system switches into an oscillatory mode.

The numerical values computed for J* and J** are presented in table 2 and depicted on the I_ext-J plane in figure 29. In addition, the quadratic differential equation of individual neurons for each region of network's activity mode are shown with the indicative colors. Dashed lines are the intrinsic functions of the neurons in absence of any input. These functions move inside the color shaded areas when an external stimulus triggers an action potential in them.

Table 2 Bifurcation points for I_ext ranging from -0.1 to -10

Figure 29. Activity regions of the network of coupled QIF neurons, plus activity regions of constituent neurons

4.6 Noise and Inverse Stochastic Resonance

To investigate the effect of noise on the life of persistent activity, the former network is turned into a system of stochastic differential equations (6), in which a white noise enters as the derivative of a standard Wiener Process, with amplitude у. Focusing on the case of I_ext = -1 and J = 6 at which we have shown that repetitive firing occurs (figure 29), some of the properties of bi-stability in presence of noise will be investigated in this section.

(6)

The two attractors of this system, stable limit cycle (SLC) and stable rest, divide the phase space in to the silent and active region. Coexistence of these two stable modes predisposes the occurrence of ISR. For a system that is in active mode, the bigger is the amplitude of noise, the more likely are the phase trajectories to escape the SCL, cross the boundary, and dislocate to the attraction domain of the rest. On the other hand, a network already in silent mode is disposed to be attracted by the limit cycle and starts oscillating due to the force of a random noise, provided the noise is strong enough to push the phase trajectories inside the attraction domain of the limit cycle. These phenomena that are shown for 4 sample trials in figure 30, are the foundations of ISR in bistable systems.

In the results presented in following the temporal output of the neurons are recorded for T = 400 ms and the noise onset time is set to 200 ms.

Figure 30. Effect of noise with four different intensities on the network of coupled neurons. Left: Phase portraits. Middle: Corresponding Spike trains. Right: Synaptic currents. a) , b) 0.1, c) 0.5, d) 1

First of all, for a noise-free condition, the neat evolution of limit cycle, regular pattern of spiking, and synaptic currents are illustrated from left to write respectively in panel a. Next, a small noise of amplitude 0.1 is injected to the network. As shown in panel b, for this case, trajectories have deviated from the SLC and after about 100 ms, they have finally absorbed by the stable equilibrium. After that, it is shown that a bigger noise of amplitude 0.5 has curtailed the firing right after the noise onset moment (panel c). It is seen in the phase plane that although, during the second half of the simulation time, noise has caused lots of movements around the S_xx, it has not become able to convey the orbits to the attraction domain of the SLC. Therefore, activities have stopped and only some small subthreshold fluctuations are observed in membrane voltage traces. Finally, a relatively large noise with amplitude 1 is applied to the system, leading to an intermittent pattern of spiking shown inn panel d. In fact, due to the great bustle, trajectories have switched back and forth between the silent and active states of the system and manifested sporadic action potentials.

In the following sections effect of noise will be studied in three conditions. First of all, J will be treated as a bifurcation parameter, while I_ext is fixated at -1 and the ISR effect will be investigated for a range of synaptic weights, below, between, and above the J* and J**. Second, J will be fixated at 15, and in a same procedure, I_ext will be treated as the bifurcation parameter. Then, ISR effect will be compared for a set of I_ext values accompanying their corresponding J*. In all of the following simulations, the mean firing rate (MFR) after the noise onset moment is acquired out of 1000 trials. The range of noise standard deviation changes from zero to 5 with a step size of 0.1.

4.6.1 J as a Bifurcation Parameter

In figure 31 the curves of MFR versus are illustrated for different fixed J values. The minimum and maximum firing rates over all trials are also shown in the shaded areas of the plots. Based on the previously founded critical points for the case of I_ext = -1 (table 2), J* = 5.8 and J** = 13. Thus, figure 31.a corresponds to the subcritical condition where the networks is still mono-stable and has only one resting state. However, it is shown here that, not only for subcritical values of J emanation of noise induced action potentials is highly probable, but also for noise amplitudes greater than 2.5 there is no avoidance from this activity.

Figure 31. ISR curves for the networks with twelve different synaptic strengths. x and y-axes show the noise intensity and the mean firing rate of neurons respectively. Boarder of the shaded areas indicate maximum and minimum of MFR. For all of the plots external current is fixated at I_ext = -1, a) J = 5, b) J J*= 6, c) J = 7, d) J = 8, e) J = 9, f) J = 10, g) J = 11, h) J = 12, I) J =J** = 13, j) J = 14, k) J = 15, l) J = 16

Besides, looking at the effect of noise where J* < J < J**, as in panels b to f, a pronounced minimum in the firing rate is observable. In fact, with increasing all the MFR curves drop at first until reaching an optimal , and then begin to raise again. These U-shape curves imply the occurrence of inverse stochastic resonance (ISR) in the network.

In addition, as seen in panels i to l for the super critical synaptic strengths J > J**, the lower bounds of the shaded areas indicating the minimum of MFR have never touched the x-axis, which means that the activity has never fully stopped. As argued before, that is because the stable node during the activity has lost its stability and there is no silent state in the phase space so that noise can deviate trajectories toward it.

Moreover, the upper bounds for all the curves, indicating the maximum firing rate within 1000 trials, suggest that even at the optimal there have been existed some trials at which the activity has not decreased but even increased due to the random perturbations.

Not all the investigated conditions of this section were shown in figure 31, but total numerical results are given in table 3 and depicted graphically in figure 32.

Figure 32. a) Optimal noise versus J. b) noise-free firing rate (blue), minimum MFR that happens at optimal noise (red), and Dif = Fr₀ - Fr_min(yellow)

The optimal noise amplitude assigned to the first column of table 3 for the subcritical J = 5 ( = 0.7) is the measured value of where noise induced spiking has emanated. Besides, the shaded cells of the table and shaded region of J-Fr plane in figure 31.b correspond to the bistable sub-region of parameter space that was marked with color pink in figure 29; The region where this system is apt to switch between its modes by random noise and thus, appropriate as a model for WM.

In figure 32.a, a positive correlation between optimal noise and J is noticeable. The greater is J, the stronger noise is required for optimally wiping out the activity. However, probability of stop at these optimal noise values are not equal for all the cases. Probability of stop by an optimal noise must be inversely proportional to the minimum of MFR (Fr_min). In figure 34.b it is seen that Fr_min is also highly correlated with J. Therefore, the weaker is the synaptic connection, the more likely it is that an optimal noise can successfully clear the persistent activity. This probability is different from the efficacy of ISR. Here, the difference between noise-free firing rate and minimum MFR, (Dif = Fr₀ - Fr_min), which seems to be an approximately bell shaped function, is proposed as a measure for ISR. This curve, shown in yellow in figure 32.b, suggests that the strongest effect of ISR occurs somewhere between J* and J**.

To explain the dependence of optimal noise on J, Gutkin et al (2004) had stated that the critical value of synaptic strength for bi-stability, here J*= 5.81, forms a limit cycle with an extremely shallow basin of attraction. Therefore, tiny jitters in the membrane voltage easily cause trajectories to escape the limit cycle toward the attraction domain of the stable fixed point, but as J increases, the basin grows in size and larger noise will be needed to deviate orbits off the attraction basin.

As it was discussed earlier, finding an attraction domain for a non-autonomous dynamical system is not possible (at least in a 2D plane), but heeding some of the phase portraits of the networks with different J values helps us understand the situations.

In figure 33 it is indicated that, in a noise-free circuit, by gradual increment of synaptic weight from J = 6 to J = 8, (panel a to c) the limit cycle is curving down and getting far from the saddle nodes. This gives more room to the orbits to be attracted by the limit cycle. In panels d and e, for J = 9 and J =11, a small I_DC emerges in synaptic currents that force the equilibriums more toward the center of the plane.

Figure 33. Noise-free phase portraits for I_ext = -1. and a) J J*= 6, b) J = 7, c) J = 8, d) J = 9, e) J = 11, f) J = J** = 13, g) J = 15, h) J = 17, i) J = 19

Where J = J** = 13, in panel f, equilibriums have converged and formed a single rest point. Therefore, for the rest of plots that have J > J**, fixed points have collided and annihilated each other (panels f to i). That is why noise has not been able to stop the activity in these conditions.

Synaptic currents corresponding to the previous plots are shown in figure 34. It is shown in the first three panels that for J = 6, 7, and 8 firing rate is relatively low and synaptic currents raise from zero level. The offset has appeared at J = 9 in panel d, and it gradually increases as a function of J from panel e to i. In panel f, that is the critical J**, DC offset is exactly equal to the absolute value of the external current (I_DC = +1). And in the last three traces (g, h, and i) the speed of activity is too high and I_DC > +1.

Figure 34. Noise-free synaptic currents for I_ext = -1 and a) J J*= 6, b) J = 7, c) J = 8, d) J = 9, e) J = 11, f) J = J** = 13, g) J = 15, h) J = 17, i) J = 19

It is now possible to grasp more explicitly why the transient excitations applied as a clear signal in section 4.3 could not extinguish persistent activity for J > 9. In fact, for J >9, when neurons are synchronized and simultaneously forced to their sub-threshold regimes, there are still some synaptic currents active in the circuit that can develop the membrane voltages and hold the activity on. In contrast to that erasure mechanism, figure 31.e suggests that random inputs with amplitude 0.5 to 1.5 may completely erase the memory and in average an optimal noise can decrees the MFR from 40 Hz to 20 Hz.

4.6.2 I_ext as a Bifurcation Parameter

In the previous section by fixating I_ext we were moving on the y-axis of the parameter space that confronted us with two critical points. Here, to check the effect of I_ext on ISR, we have, first fixated J at 15 and repeated the same procedure of measuring ISR for different values of external current. Inquiring figure 29 again, we figure out that, moving on this axis, we will encounter three critical conditions. However, since it is known that for I_ext > 0, the individual neurons are intrinsically oscillatory, we have only considered the excitable network for which neurons have non-positive external currents, ranging from -4.5 to 0. The ISR curves obtained out of these simulations are indicated in figure 35.

In panel a, where external current of size -4.5 is not sufficient to switch on the activity in the noise-free condition, the system is still mono-stable and neurons are at rest. However, noise induced activity has emerged and grown as a function of . In panel b, with a little bigger external current I_ext = -4 the bifurcation has occurred and the network has entered a new bi-stable mode. The exact bifurcation point for bi-stability is I_ext* = -4.1. Here again, the U-shape function of MFR versus noise is a clear indicator for the occurrence of ISR.

Similar to what was observed in the previous case, as the system gets farther from the bifurcation point, I_ext*, probability of stop decreases gradually. This fact is easily apprehensible by following the pattern of ISR curves from panel c to i. (The closer is the minimum of MFR to zero, the more likely it is that noise can stop the activity). In panel g, the system is almost on the boarder of the second critical external current, I_ext** = -1.49. As it is seen for less negative external currents, I_ext > I_ext**, in panels h, and i, noise has almost no effect on MFR, and the minimum of MFR is always greater than zero, i.e. the activity has never stopped.

Figure 35. ISR curves for the networks with twelve different external currents. x and y-axes show the noise intensity and the mean firing rate of neurons respectively. Boarder of the shaded areas indicate maximum and minimum of MFR. For all of the plots synaptic weight is fixated at J = 15, a) I_ext = -4.5, b) I_ext I_ext* = -4 c) I_ext = -3.5, d) I_ext = -3, e) I_ext = -2.5, f) I_ext = -2, g) I_ext I_ext** = -1.5, h) I_ext = -1, i) I_ext = -0.5

Numerical results for these simulations are given in table 4 and illustrated graphically in figure 36. The optimal noise amplitude assigned to the first column of table 4 for the subcritical I_ext = -4.5 () is the measured value of where noise induced spiking has emanated. As expected and also indicated in panel b the noise-free firing rate and I_ext are positively correlated with each other. Fr_min grows with I_ext that implies the reduction of stopping probability, while the Dif function is maximum between I_ext* and I_ext**.

Table 4Numerical properties of ISR for fixed J = 15

Figure 36. a) Optimal noise versus I_ext. b) Noise-free firing rate (blue), Minimum MFR that happens at optimal noise (red), and Dif = Fr₀ - Fr_min (yellow).

The illustrated data in panel a of this figure suggest that there is a positive correlation between the optimal noise intensity and I_ext, i.e. the less negative is the external current the smaller noise is needed to optimally extinguish the firing activity. This is not surprising, since by increment of I_ext neurons are approaching their oscillatory mode and their activity become more and more robust.

To explain the dynamical properties of the system, some of the phase portraits and the synaptic current patterns for the noise-free networks with fixed J and different external currents are plotted in figures 37 and 38 respectively. For the external current of size -4 that is very close to the bifurcation point I_ext**, the new born limit cycle is shown in figure 37.a. It is seen that the SLC is almost abutting the saddles, so it must have an extremely shallow attraction domain. The corresponding synaptic currents shown in figure 38.a indicate that the firing is still slow enough so that exponentially decaying synaptic currents reach and raise from zero level.

Figure 37. Noise-free phase portraits. For all the networks J = 15. and a) I_ext I_ext*= -4, b) I_ext = -3, c) I_ext = -2, d) I_ext I_ext** = -1.5 , e) I_ext = -1, f) I_ext = -0.5

As a result of less negative external current, in panels b and c of both figures, equilibriums have approached each other. Besides, I_DC that emerges at I_ext = -3.5 applies an additive force to the equilibriums and Push them more toward the center of the phase plane. Panels d of these figures show the condition of saddle node bifurcation during activity, where there is a single degenerate rest point in the phase space and DC offset is equal to the absolute value of I_ext. Meanwhile, by growth of I_syn from I_DC this single node collapses into four complex equilibriums and returns back in the decay period of I_syn. And finally in panels e and f, fixed points have already vanished during the activity. The hollow circles in figure 37 indicate the location of intrinsic equilibriums that are not present during persistent firing.