PROCEDIMIENTO PARA EL ANÁLISIS DE INFORMACIÓN

To understand the qualitative dynamics in absence of an analytical solution we turn to the tools of dynamical systems theory (Strogatz, 1994). We first take a look at trajectories of the discrete map in (m, n) phase space. The trajectory bundles are best understood by examining the nullclines of the system, which are the loci of points where one further iteration does not change at least one of the coordinate e.g. mt−TOn(mt, nt) = 0. We finally summarise the region of pattern sizes M and

thresholdsθ where replay is dynamically successful in a phase diagram, aθ(M)plot generalizing in the mean-field the classification of dynamical regimes first seen in Figure 4.8. The phase diagram allows to explore the factors that affect the min- imum pattern size Mopt and thus the capacity of the system for sequence memory (see Eqs. 4.12 and 4.13).

4.3.3.1 Trajectories in the Phase Plane

In the forthcoming sections we shall be discussing the attractors of the dynamics of the mean-field map. The underlying cellular dynamics does not, however, have a fixed point and does not thus fall in the category of attractor memory networks (Hopfield, 1982). For the representation to be informative, we have chosen to dis- play phase space in scaled coordinates (n/Fvs.m/M), and only its lower-right corner

close to the point of perfect replay (see green box in Fig. 4.7). A nonlinear scaling would distort the trajectories, and once trajectories escape the lower-right corner of phase-space they go straight to the all-silent fixed point at lower left or to the all- active one at upper right. Finally, in this representation, the displacements in both directions give rise to equivalent changes in the replay quality (Eq. 4.11).

4.3.3.2 Nullclines and Fixed Points

The nullclines of a phase portrait are the points where the flow is exclusively horizontal (n-nullcline, given in implicit form bynt −TOff(mt, nt) = 0) or vertical (m-nullcline, mt−TOn(mt, nt) = 0). They break up phase space in regions according to the sign of

the change inmandn. Nullclines do not tell directly about the stability of the regions they demarcate because they can be crossed by the flow, but their crossings are the fixed point of the system, where the flow comes to a halt (mt+1−mt= 0;nt+1−nt=0).

n nullcline for high n are shown separately in two insets. The lower-right corner is

zoomed-in (yellow) to facilitate analysis of the fixed point of retrieval.

Fixed points The crossings of the nullclines reveal three stable fixed points close to

(m/M , n/F)= (0,0)(all-silent),(1,1)(all active) and(1,0)(retrieval), in this example withM= 1,000, θ=79. These fixed points are indicated by discs in Figure 4.14. The remaining fixed points are saddle points (repulsive in one direction and attractive in the other), and are indicated by squares.

0 20 40 60 80 100 Hits m / M (%) Fa lse a la rm s n / F ( %) 0 0.5 1 1.5 2 -- 9,700 -- 9,100 m / M (%) 92 96 n / F (%) 96 100 95 100 n / F (%) 100 m / M (%) 0.1 0 96

Figure 4.14. Overview of Nullclines and Fixed Points of the Mean-Field Model. The n-nullcline is shown dashed (mirror-Z-shaped); the m-nullcline full (N-shaped). Nonphys- ical sections of the nullclines withm/Morn/F_∈[0,1]are greyed out. Arrowheads indicate the repulsive (brown) or attractive (green) character of each of the nullcline branches. The main plot (bottom panel) shows the whole range ofm/M _∈[0,100%]and the range ofn/ F <2%that is of interest for retrieval dynamics. The turning point of them-nullcline is shown at left; the all-active fixed point is shown in the inset at top. The yellow region zooms into the area of replay. It exhibits a saddle fixed point (square) and the stable fixed point (disc) of high-quality retrieval. Parameters wereM= 1,000, θ=79, c=0.05, cm=0.1,andN=105.

Shape of the nullclines The shape of the nullclines can be understood by referring to

the sigmoid function(1 +erf(z))/2(depicted in Fig. 4.15). The sigmoid is centered

onz= 0. It is mostly flat and close to zero for z <₋2, mostly linear for −2< z <2

and mostly flat and close to 1 for largerz(this derives from the fact that two standard

deviations about the mean contain about 95% of the area under a Gaussian curve). In our case, the scaling ofzis provided by the standard deviation of the synaptic input

to the respective population,σOn/Off, and its originz= 0by the difference of the mean

inputµOn/Offwith the thresholdθ. Let us discuss for example themnullcline (black line

in 4.14) and disregard for the moment the inhomogeneity inm, nof the scalingσOn: m_t= [1 +erf((c_mm_t+c n_t−θ)/σOn)]/2.

Formt≃0 (mnullcline) there will be a range of nt such that (c nt −θ)/σOn<−1. In the Figure that range stretches until aboutnt/F=0.012 ornt≃1,200. In view of c=0.05 andθ=80 we have(60−80).₋σOn. NowσOnin the region of phase space

m_≃0, n= 1,200 is about 9 (Table 4.7). We see that the qualitative conditionz.₋2is

fulfilled. Once we reach that point the linear section of the sigmoid kicks in (Fig. 4.15). A tradeoff between activation coming from hits and false alarms governs this linear section of the nullcline. There mt∝(cmmt+c nt−θ)/σOn. The negative slope cor-

responds to the increasing compensation of false alarms by hits as the number of the latter grows. Eventually, the region wheremt ≃M is reached. If the threshold had

been tuned to support replay, it should be slightly lower thancmM so that a majority

of On neurons fire, and the nullcline will reach almostmt=M , nt= 0. From there on,

further activation of Off neurons at maximum hit activity is already so much above threshold(zOn>2, σOn(M ,0)≃10)that them-nullcline overlaps completely with the

mt=M vertical line. 1 0 0.5 0 -- 2 -- 4 2 4

Figure 4.15. The Sigmoid Function Has Three Roughly Linear Regimes. The cumulative

distribution function of the normal is a shifted and scaled error function. It is the prototypal threshold nonlinearity (sigmoid), with three regions that can, each, be approximated linearly. The leftmost regionz_∈(_−∞,₋2)is the under threshold region. Regardless ofzthe value of the function is very nearly zero. The central regionz_∈[₋2,2]is approximately linear with slope1/ 2√ p(21.75◦). The rightmost region is the saturation region. Regardless ofz∈(2,∞) the value of the function is very nearly 1. The region limits₋2,2have been set arbitrarily for illustration; they contain 95% of the area of the Gaussian. The intersections of the central straight line with 0 and 1 are at_±√p/2≃1.25.

A similar reasoning can be followed for then-nullcline although the quantitative details are different due to the sparsity of the network (M _≪ F) and the different connectivity (cm= 2c). Importantly, the connectivities influences the slopes of the

diagonal section of the nullclines, thus controlling whether they cross to generate the fixed point of stable replay.

Effect of threshold on the phase plane The nullclines of the mean-field dynamical

system are shown in Figure 4.16 B for a threshold setting that, similar to similar to Figure 4.14, allows for stable retrieval: there exists an asymptotically stable fixed point

(m_∞, n_∞) = [TOn(m∞, n∞), TOff(m∞, n∞)]

95% 100% Hits m / M .4% 0% Hits m / M 95% 100% .4% 0% 95% 100% Hits m / M Fa lse al ar m s n / F 122 q = q =127 q =135 A B C

Figure 4.16. Effect of the Firing Threshold on the Nullclines and Fixed Points. B. The

mnullcline (solid line) intersects twice with thennullcline (dashed), producing stable (disc) and unstable (square) fixed points. Arrows indicate attractive or repulsive character of the nullcline; gray areas correspond to unphysical valuesn <0orm >M.A. Threshold too low; no

fixed points. The trajectories starting out nearby a perfect pattern escape to the all-active fixed point at(m/M , n/F) = (1,1).C. Threshold too high. Only the unstable fixed point remains;

trajectories escape to the all-silent fixed point at(0,0).

If the firing threshold is too low or the pattern size is too large (Fig. 4.16 A), the nullclines do not cross in a retrieval regime: After initialization at the condition of perfect retrieval(m0, n0)=(M ,0), all neurons immediately start to fire and the network

falls into an all-active state,(m, n)≃(M , F). If the firing threshold is too high or the

pattern size is too low (Fig. 4.16 C), only an unstable fixed point exists in the retrieval region. After initialization at perfect retrieval, the network immediately falls into an all-silent state(m, n)≃(0,0).

Note that in Figure 4.16 B as compared to Figure 4.14 (yellow inset) themnull- cline intersects the horizontal branch of then nullcline atn= 0. This fact allows to

apply the implicit function theorem to the nullclines so as to extract proper functions

m(n)and find out in closed form whether they intersect atn= 0. These formulas will work unfortunately only for the case where the loss of stability is, as depicted in C, by too high a threshold, because the curvature of the branch of thennullclinen(m)in the vicinity ofm=M is not negligible. The original idea of applying the implicit function

theorem is due to Axel Kammerer, and we have sketched it here as a possible starting point for further analytical work, since it can provide a good approximation of the upper edge of the stability wedge (cf. next Section) for many networks of interest.

4.3.3.3 Phase Diagram: the Wedge of Stability

In this Section we investigate systematically which combinations ofM , θenable high- quality, sustained replay. For that, we employ aphase diagramthat displays the asymp- totic stability of a range ofM , θ combinations by a color code (choropleth map).

This phase diagram (Fig. 4.17) reveals the three phases of a sequence memory network that we have come to expect from cellular simulations: all silent, all active, and retrieval. The region in which retrieval is possible is wedge-shaped with a thin tip at low pattern sizesM. We tested the region of retrieval obtained from the mean- field equations with extensive computer simulations of the corresponding networks of binary neurons (white discs in Fig. 4.17; see Section 4.1.6). As expected, owing to the finite size of the simulated network, the region of retrieval is overestimated by mean-field theory, yet the deviations are relatively small. According to Eq. 4.12, the numberPof stored associations increases with decreasing coding ratiof =M/N, and thus the network displays the highest memory capacity at the wedge tip M =Mopt.

There the stability of the fixed point is particularly sensitive to noise and thus the high capacity is not accessible unless the dynamics can be stabilized. This could be done by a mechanism that senses the escape of trajectories to the non-replay fixed points and adapts the threshold accordingly. We therefore investigated whether inhibition can provide such stabilization by effectively adapting the threshold.

Pattern size M 1,600 2,000 Th resh ol d B all silent all active A ret riev al C M = 1 ,6 0 0 150 100 50 125 75 800 1,200

Figure 4.17. Phase Diagram.M-θspace consists of three areas:All silent; the sequence dies out.All active; all neurons fire at maximum rate.Sequence retrieval(black); the fraction of hits is much larger than the fraction of false alarms for infinitely many time steps. The diagonal dashed line at bottom left separates the all-silent and all-active phases forM-values at which no retrieval phase exists. Areas in light gray correspond to transient retrieval of at least 3 time steps. White discs mark the boundary of the retrieval region as obtained from simulations of N binary neurons for exemplary values ofM. Note colored labelsA,B,Cmarking theM , θ combinations of the respective phase-space panels in Figure 4.16. Parameters areN=105_,

M= 1,600,cm=0.1, andc=0.05.