Modelos y concepciones alternativos en evaluación de los aprendizajes de los estudiantes

As a coda for our investigation of inhibitory feedback we briefly and unsystematically explore the impact of recurrently coupling the inhibitory pool. Anatomical and elec- trophysiological investigations in the hippocampus show that inhibitory interneurons project to dendritic shafts of other interneurons. An inhibitory population that is cou- pled to itself is is governed by the dynamical equation (compare Eq. 4.35):

kt+1 K = 1 2 " 1 +erf cEIwEI(mt+nt)−cIIwIIkt−η 21/2 wEI2 cEI(1−cEI) (mt+nt) +wII2cII(1−cII)kt p !# . (4.37) r,c color K/103 wIE/10−3 cEI wEI cII wII η note

1,1 blue 5 12 0.01 1 — — 8.8 Fig. 4.22; cyan:k0=K/2.

1,2 green 20 4 0.0085 1 0.0003 1 4 light green:k0=K/2;Mopt&700.

1,3 brown 58 0.8 0.5 1 0.5 0.01 50

2,1 black 58 0.8 0.5 1 0.65 0.007 45 Mopt≃700.

2,2 pale blue 20 4 0.001 1 0.0005 1 4 k0=K/2;Mopt&700.

2,3 orange 50 0.8 0.5 1.3 0.5 0.008 600 broken wedge.

3,1 red 80 0.5 0.5 1 0.5 0.006 180

3,2 purple 64 0.86 0.52 1 0.94 0.0048 44 Mopt&700.

3,3 navy 28 1.1 0.4 0.4 0.45 0.002 120 broken wedge.

Table 4.8. Parameters Used for the Exploration of Recurrent Inhibition. Position in the

grid of Fig. 4.24 is indicated as row, column (r,c). All networks have in commonN=105_,

c=0.05,cm=0.1, andcIE= 1, and were initialized withk0=TInh(M ,0,0)unless noted.

Note that finding a fully self-consistent initialization is harder to find because of the direct dependency of kt+1 on kt. We decided to compare two approaches in

Figure 4.24, where we show the regions of stability for different combinations of parameters. One, as in the previous Section, assumes that the starting activity of the inhibitory pool is consistent with the activation of M excitatory hit neurons only,

as corresponds to pattern retrieval, and an initially silent inhibitory population, i.e.

k0/K=TInh(M ,0,0). The other consists in setting k0=K/2, and without recurrent

inhibition often leads to disintegration of the whole wedge. However, recurrent inhi- bition is able to sustain an asymptotic state of replay in an islandat low pattern sizes (see green and pale blue wedges in Fig. 4.24). Remarkably, there exist parameter combinations for which a network with recurrent inhibition slightly improves on the capacity achieved without it, but finding them requires patient exploration. Armed with the intuitions of Eq. 4.4.3 about the relationships between the three weightswEI,

wIE, wII, the three corresponding connectivities, the inhibitory thresholdη, the number of inhibitory neuronsK and the initializationk0, it is possible to reduceMoptby about

10%, as shown by the saturated-color wedge tips and verified by calculating indi- vidual trajectories e.g. on the light green and light blue wedges. Table 4.8 shows the parameter values used for the Figure. Based on our sampling of parameter space, which is no substitute for a more systematic optimization approach or proper ana- lytical bounds, it seems that recurrent inhibition can help robustness at the tip and push the stable region to slightly lowerMopt. It is however unlikely that it can operate

higher gains than e.g. theMopt&700 achieved in some of the examples below.

Figure 4.24. Phase Diagram Examples in Presence of Recurrent Inhibition. Nine net-

works with recurrent inhibition (Table 4.8) are compared to a network wherecII= 0(blue, top left, identical to blue in Fig. 4.22). Initialization withk0=K/2is demonstrated in lighter color

for the first two networks and in the central example for both colors; all others are initialized withk0=K TInh(M ,0,0). A blue vertical line marks pattern sizeM=890, the approximate tip of the stable region without recurrent inhibition in the top-left panel.

Part IV

Discussion

Chapter 5

Ripples, Assemblies, and Sequences

In this final Chapter we collect our results and discuss them in the light of the present state of knowledge. We also take the liberty to suggest future lines of work. Figure 5.1 presents the unifying view of the two strands of this Thesis: the descriptive of signal analysis of noisy electrophysiological data, and the prescriptive of a simplified math- ematical network model. These are bound together by the concept of neuronal assem- blies, that make up the memory traces and the heartbeat of the ripple.

Network Capacity for Sequence Replay

modeling: mean field dynamics

data analysis: peeling deconvolution Biophysical Mechanisms of Sharp-Wave Ripples SWR- associated intracellular currents network observers 1 2 3 4 5 6 7 cycles / patterns cu rren ts pres yn ap ti c a ct ivit y ne uron s pres yn ap ti c s pike s 1 2 3 4 5 6 7 postsynaptic currents recordings in CA1 mouse pyr Maier et. al, Schmitz lab

Figure 5.1. Summary of Research Presented in this Thesis. We used as a starting point currents measured in voltage clamp at CA1 pyramidal neurons (center, black line enclosing grey area) concurrent to sharp-wave ripples in the LFP (not shown) as recorded from a mouse slice model (IRDIC image at left, cell5is patched). Analysis of the composition of the cur- rents (top) revealed a rhythmic succession of EPSCs (colored curves; discs below sized by amplitude) and IPSCs (not shown). The notable modulation of pyramidal cell activity in a 5 ms rhythm together with literature reports of replay of behavioral sequences during ripples inspired a theoretical model of capacity for sequence replay based on coactivating assemblies at each ripple cycle (bottom, cycle-discrete schematic raster plot).

In the first part (upper panel; dark red) we studied currents incoming to CA1 pyramidal cells during SWR in vitro. We used a novel peeling reconstruction approach in combination with more traditional signal analysis techniques to obtain a quantitative characterization of the contributing excitatory and inhibitory PSCs, including their rhythmicity, cycle-dependent amplitude and relative timing.

In the second part (lower panel; dark blue), inspired by the above results on ripple locking of currents and numerous reports that SWR support the retrieval of behavioural spatial sequences, we built a mean-field model of a hippocampal recur- rent network with inhibition. With it we could obtain the optimal number of neurons active in an assembly and what their threshold should be so as to concurrently opti- mize capacity and robustness for the replay of sequences.