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

Moreover, looking at the structure of phase planes in figure 37, it is seen that by increment of I_ext the saddle nodes are distancing from the limit cycle. This fact explains the spatial growth of the SLC's attraction domain and the growth of required noise for termination of activity.

Figure 38. Noise-free synaptic currents. 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

4.6.3 Bifurcation with different parameters

In this section, ISR properties are investigated for different pairs of J*-I_ext. All the networks are at their global bifurcation points, with extremely fragile limit cycles, but their noise-free firing rates are varying with the alternative pairs of parameters. ISR functions for some of these cases are shown in figure 39. What is firstly notable in these figures is the similar steep slopes of curves from Fr₀ for all the plots. In all of these cases MFR has soon extinguished by tiny perturbations, trivially, because all the systems are at critical conditions.

Figure 39. ISR curves for networks with nine different pairs if J*-I_ext. 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. a) I_ext = -0.1, J* = 1.48, b) I_ext = -0.5, J* = 3.77, c) I_ext = -1, J* = 5.9, d) I_ext = -2, J* = 9.2, e) I_ext = -3, J* = 12.1, f) I_ext = -4, J* = 14.8, g) I_ext = -6, J* = 19.7, h) I_ext = -7, J* = 22, i) I_ext = -8, J* = 24.3

Optimal values of , Fr₀, and Fr_min for 21 different pair of parameters are presented in table 5. These numerical data are graphically depicted in figure 40 as well. In this figure x-axis corresponds to the values of I_ext, but it should be noted that in accordance with increment of the external current, the critical synaptic weight, J*, is decreasing. These results signify that Fr_min, that is the probability of stop, is almost constant for all the networks at their critical condition, no matter what is the speed and period of SLC. In other words, while coupled neurons are close to global bifurcation, an optimal noise always exists which can easily wipe out the persistent activity with an almost constant expectancy.

Table 5Numerical properties of ISR for different pairs of J*-I_ext

Figure 40. 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 pattern of optimal noise values for these range of parameters in figure 40.a seems strange and difficult to be explained thoroughly. This factor is supposed to vary relative to the size of SLC's attraction domain. Although, all of the limit cycles as shown in figure 41 abut the saddles, the fluctuating pattern of optimal noise suggests that the size of attraction domain is not equal for all of them.

Figure 41. Noise-free phase portraits for bifurcation points with different parameters. a) I_ext = -0.1, J* = 1.48, b) I_ext = -0.5, J* = 3.77, c) I_ext = -1, J* = 5.9, d) I_ext = -2, J* = 9.2, e) I_ext = -3, J* = 12.1, f) I_ext = -4, J* = 14.8, g) I_ext = -6, J* = 19.7, h) I_ext = -8, J* = 24.3, i) I_ext = -10, J* = 28.6

Besides, the corresponding synaptic currents for the networks close to bifurcation are also shown in figure 42 indicating that synaptic time constants are relatively fast for all of the discussed networks. Therefore, there are only two changing variables in the phase space for different pairs of parameters which might have affected the size of attraction domain and optimal noise; (1) First, the amount of time that equilibriums spend far from their intrinsic locations by the force of J; (2) and second the distance they travel on the plane depending on the strength of the synapse. However, it is not clear that how these factors have shaped the pattern of optimal noise in different conditions. Of course, time dependency of the system under discussion makes the problem more complicated.

Figure 42. Noise-free Synaptic currents for bifurcation points with different parameters. a) I_ext = -0.1, J* = 1.48, b) I_ext = -0.5, J* = 3.77, c) I_ext = -1, J* = 5.9, d) I_ext = -2, J* = 9.2, e) I_ext = -3, J* = 12.1, f) I_ext = -4, J* = 14.8, g) I_ext = -6, J* = 19.7, h) I_ext = -8, J* = 24.3, i) I_ext = -10, J* = 28.6

4.6.3.1 Concavity Width

Another appreciable feature in figure 39 that had not been discussed in any of previous studies, is the growth of concavity width of ISR curves from panel a to i. The wider is this width, the more difficult is transition from silent to the active state. We will mark this factor by the name `CW' in this thesis.

Based on the explanations of Uzuntarla et al (2013) about the 3 key features of a dynamical structure leading to occurrence of ISR, it might be hypothesized that the farther is SLC from the rest point, the more difficult would be to move from rest to active state, and thus the bigger should be CW. However, inquiring the noise-free phase portraits presented in figures 41 disproves this hypothesis. It is shown here that as external current goes more negative, SLC comes closer to the rest point. Nevertheless, we saw that for more negative I_ext concavity width is wider.

In section 4.2.1 we argued that the required J for switching on the persistent activity depends on I_ext. With the same analogy we suggest that the required energy for moving from silent to active mode by noise, also depends on the force of external current. Therefore, it is proposed that CW must be a direct function of I_ext, or in other words, a function of the distance between stable and saddle nodes. The farther are these two equilibriums, the bigger is CW.

Presumably, for an appropriate model of WM that take the advantage of randomness for erasing the irrelevant information, it is better to stay in a range of parameters which not only provides bi-stability and closure to a global bifurcation point, but also provide a relatively large CW. This characteristic assures us that random noise doesn't produce irregular and nonsense firing activity in the model.

4.7 Coloured Noise

Since realistic background noise in biological neural circuits is reported to be more colorful rather than white (Destexhe & Contreras, 2006), effect of coloured noise is also shortly investigated with the same procedure as before. A first order autoregressive (AR) filter with the transfer function is utilized for producing coloured noise. A set of Gaussian white noise ranging from to , are filtered with four different values of . parameter in this function defines the level of autocorrelation for the output noise. ISR effect is investigated in several networks having fixed external current, Iext = -1, and J values ranging from 5 to 13. Results as shown in figure 43 suggesting that everything is almost similar to the case of white noise except the values of optimal noise that shift more and more to the right as a function of the noise color.

Figure 43. ISR curves for 4 different coloured noise and 9 values of J. Color of noise is indicated by the colors of curves. = -0.6 (blue), = -0.7 (red), = -0.8 (yellow), = -0.9 (Purple), for all the panels Iext = -1. a) J = 6, b) J = 7, c) J = 8, d) J = 9, e) J = 10, f) J = 11, g) J = 11, h) J = 12, i) J = 13.

Numerical results are given in table 6 and illustrated in figure 44. It is seen that ISR efficacy measured by Dif has the same bell shape function that it had for the white noise. Probability of stop with optimal noise doesn't change significantly for different colors of noise, and the optimal value of noise has a positive correlation with J. What is important to notice is the magnitude of this correlation. The greater is the less correlation is observed between J and optimal noise intensity, insofar that for = 0.9 optimal noise is almost independent from synaptic weight.

Table 6 Numerical properties of ISR for four different coloured noise and fixed J = 15

Figure 44. 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).

4.8 Left and Right Hand Side of ISR

It was shown that by increment of noise intensity, MFR falls down, reaches a minimum, and grows again to overtake Fr₀. Therefore, there are pairs of on the left and right hand side of the ISR curve provoking quantitatively equal mean firing rates. But, are the quality of firing activities also similar at these points? To answer this question, first of all, the MFR for a typical network with J = 6 and I_ext = -1 is again acquired as a function of noise amplitude. This time the simulation time after noise onset is taken longer (800 ms) so that we can observe the spiking patterns more clearly. All the other parameters are the same as before and the obtained ISR function is depicted in figure 45.

Figure 45. MFR versus noise intensity during 1 s after applying noise.

Based on the data given by this curve it is expected that the mean firing rate at = 0.85 and =0.1 are equal. In figure 46 we have plotted the spike trains of 10 sample trials for each case.

Figure 46. 10 sample trials of spike trains produced as a result of applying random perturbations. a) = -0.1, b) = -0.85, The last trace in both columns show the random noise coming at t = 200 ms at remains on, for 800 ms.

Column A corresponds to the left side of the ISR curve with small noise of amplitude 0.1 and column B Corresponds to the right hand side with a bigger noise of size 0.85. Obviously, small noise has soon or late switched off the system, but it has never become able to turn it back from silent to active mode. In contrast, large noise has almost always switched off the activity as soon as its arrival, but it has caused the system to move back and forth between its two stable modes, generating an intermittent pattern of firing.

4.8.1 Firing rate distribution

Distribution of Firing Rates is an appropriate measure for indicating the qualitative differences of firing activity. We have found histograms of firing rate for four values of noise intensity that are expected to enforce similar MFRs in the network. Based on the ISR curve in figure 45, the system exposed to a noise of size = 1.75 exhibits the same firing rate as a noise-free system. These two conditions are compared in panel a and b of figure 47 by showing the distribution of their Mean Firing Rates. Not surprisingly distribution of Fr₀ stays constantly on 21, but on the other half of the ISR curve, although mean firing rate is still about 21, the distribution has a wide Gaussian form.

Figure 47. MFR histograms. a) Noise-free, b) = 1.75, c) = 0.1, d) = 0.85

In addition, firing rate histograms for intermediate noise levels on the left and right side of ISR curve are compared in panel c and d of the same figure. Both histograms are skewed to the right and have equal mean values around 2.5. However, the peak on zero is larger for = 0.1 and skewness is greater for = 0.85. It signifies that, although MFRs are equal, having no spike after noise onset is more probable with the small noise on the left, than with the larger noise on the right.

4.8.2 Inter Spike Interval Distribution

Another tool for studying ISR is the histogram of inter spike intervals (ISIH). It is known that on the right side of optimal noise, spike trains become more irregular and of intermittent type, while on the left side activity is much more regular. Consequently, ISIH patterns as indicated in figure 48 are wide on the right (panel b and d) and narrow on the left (panel a and c). In addition, for very large noise intensity, The speed of firing will be so high that leads to decrement of ISI variability. That is why ISIH for = 1.75 (panel b) is more contracted than = 0.85 (panel d).

Figure 48. Inter Spike Interval Histograms. a) Noise-free, b) = 1.75, c) = 0.1, d) = 0.85

Chapter 5. Discussion and conclusion

A successful model of WM in cellular level, should, firstly, be able to store information by generating a regular spiking activity in reaction to an external stimulus, and sustain this activity when that stimulus is removed. Second, it must be able to execute an erasure mechanism by quenching the mnemonic activity within a short time. To implement these mechanisms, we need a bistable system disposed to switch between an oscillatory state and a resting state. For this purpose, we have introduced a minimal model comprising 2 QIF neurons that reciprocally excite each other and form a kind of neural oscillator, simulating the persistent activity of cortical delay selective neurons in a WM task.

Each neuron is a canonical type 1 model that can go through a saddle node bifurcation from an excitable mode in to a periodic mode of activity. The single neuron can be whether excitable or oscillatory at one time, but it was shown that when a pair of these neurons are coupled by a sufficiently strong synapse, intrinsically excitable constituent cells behave periodically due to the periodic force of a synaptic current that shifts their equilibriums back and forth in the phase space. Therefore, a stable limit cycle' corresponding to the periodic activity, and a stable rest node determined by the fixed points of the individual cells, shape the two attractors of this proposed bi-stable system. In addition, it was shown that the closer (farther) are the individual cells to their intrinsic bifurcation, behaviour of the circuit looks more type 1(2).

Since spiking activity in the recurrent networks are always asynchronous, a transient signal applied to both cells simultaneously increases the level of synchrony and ends up to abrupt termination of activity. Such a transient signal might be provided by a reward or by a motor command after accomplishment of a task, or just by the adjacent response selective cells that are discovered in PFC. Also, the clearing signal might be of excitatory or inhibitory type, but it was shown that excitation doesn't work in the case of strong coupling.

This finding is a direct consequence of an exponential time dependent synaptic kernel that was introduced to the model for making it more biologically realistic. The previous minimal models proposed for maintaining a memory by Novikov and Gutkin (2016) had employed voltage dependent synapses that could only explain the condition of limited range of parameters where the synaptic time constant were relatively fast. Moreover, it is suggested that since synchronizing neurons of a circuit having autonomous synapses simultaneously leads to synchronization of the synaptic currents, operation of these models might not be trustable for the conditions of strong synchrony and neither for the situations of strong coupling. When the synaptic strengths connecting these the cells is strong the speed of activity will grow up in a way that its dynamical structure and stability properties change. Therefore, the non-autonomous version of this model studied here can simulate properly operate for a broader range of parameters.

In the present thesis dynamical features of the non-autonomous minimal model of WM were investigated in a 2D parameter space. First, external current that is intrinsic to individual cells was fixated, and by orderly increasing synaptic weights, two critical points were found for the network. (1) One global bifurcation point at which the excitable mono-stable system turns into bi-stability and a stable limit cycle emerges in the phase space, coexisting with the former rest point; (2) And another critical point at which the whole phase plane is covered by the attraction domain of the limit cycle in a short transient period after it is switched on. The latter case happens when the firing rate grows too high and synaptic time constants become relatively slow. It was shown that although the system is still considered bi-stable after the second critical point, the activity is not stoppable by transient excitation and nor by a random noise. The only clearing mechanism functioning for this case is application of a transient inhibitory signal that can decrees the synaptic currents and returns the stable equilibrium back to the phase plane.

In another attempt, we have fixated J, and walked along the external current-axis in the phase plane. It is shown that as a result of varying I_ext, the system encounters 3 critical points. The first and second points are similar to the previous case, while there is a third local bifurcation point at zero external current that conveys the single cells across their saddle node. After this third point the system is mono-stable again, with a single attractor that is the stable limit cycle. i.e. even in absence of any input or any coupling neurons are oscillatory and always fire for I_ext > 0.

We introduced a sub-region in the parameter space that provide reasonable bi-stability appropriate for modelling `maintenance' and `clear' operators of the WM. In addition, considering another operation `prevent' we suggest that it is safer not to go very close to the global bifurcation point. Because at this point the persistent activity is too fragile and does not sustain even in presence of tiny distractors.

The effect of random perturbations was also studied on the activity of our intended model. In particular, the ISR phenomenon was investigated in different conditions. It was shown that between the two critical points, where the system is bi-stable, a random noise can deviate the phase trajectories from the limit cycle and dislocate them in the attraction domain of the stable rest. this phenomenon is proposed as another clearing mechanism which might be performed by biological neurons to taking advantage of randomness in background activity to extinguish the irrelevant mnemonic firing.

By numerical computation of mean firing rate versus noise intensity we have proved that the probability of stop increases by increment of noise, reaches a maximum at an optimal noise level, and decreases again for stronger noise. This pattern is the first indicator of ISR in nonlinear dynamical systems that was proved to occur in our model both for white and coloured noise.

in addition, the larger is the attraction domain of a limit cycle the less probable is that phase variables can escape it. Therefore, as the attraction domain of the limit cycle grows in size probability of stopping the activity decreases. The attraction domain itself is directly proportional to J (I_ext), while I_ext (J) is constant.

Moreover, we have shown that the probability of stop is different from ISR efficacy. To measure the ISR effect we have calculated the difference of noise-free firing rate and the minimum achievable mean firing rate in presence of noise in different conditions. Our results have shown that this function, called Dif, always reaches a maximum somewhere between the two critical points.

The optimal value of noise determines the steep of decay from Fr₀ in ISR curves. This factor is positively correlated with the size of attraction domain. The larger is the attraction basin of the limit cycle the more energy is required to push the orbits out of it. On the other hand, the Concavity width (CW) of the ISR curves depends on how much energy is required for escaping the stable rest state to the limit cycle. We showed that CW is a function of I_ext. The more negative is I_ext the wider is CW.

Distribution of firing rates and ISI histogram were also presented as two measures for ISR. Inquiring these measures, it must be noticed that if the information is encoded in firing rates of such bistable networks, a particular noise can induce the same firing rate as the noise-free system exhibit. This may mislead the information decoding in presence of noise. In such conditions the CW must be chosen wide enough to ensure reliability.

However, it doesn't mean that great levels of CW are always advantageous for any system. We believe that this minimal model is generalizable to many nonlinear dynamical systems that have accompanying stable limit cycle and stable rest in terms of ISR. By modifying parameters of such systems and narrowing down CW, the ISR phenomenon can be used as a filtering mechanism for damping a particular noise.

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