En busca de la transformación de las prácticas evaluativas, hacia un enfoque crítico y

Neural network implementations of associative memory have been profusely studied, first only with excitation and dense connectivity (Willshaw et al., 1969), then intro- ducing sparse connectivity (Gibson and Robinson, 1991) and eventually adding global feedback inhibition, e.g. Golomb et al. (1990), Hirase and Recce (1996). However, in real neural networks interneurons do much more than instantaneously adjust the gain by controlling the threshold of excitatory neurons. They also display abundant recur- rent connectivity (Amitai et al., 2002; Yuste, 2011), so that it is useful to consider them as a population and study their behaviour in time beyond threshold scaling.

Here, combining cellular simulations with a mean-field model, we studied large, sparse associative memory networks inspired in hippocampus in order to assess sequence replay in the presence of inhibitio. We first addressed global instantaneous inhibition and asked what would be the feedback coupling function that optimizes capacity. We next introduced a full inhibitory population, on an equal footing with the excitatory population, and studied how its activity evolved during the replay phenomenon. We finally considered self-coupling in the inhibitory population and explored its functional impact on the capacity and robustness of sequence replay.

5.2.1 Executive Summary

Here are in short form the methods that we used to address the question of optimizing memory capacity for sequences, the results we obtained and some of the further ques- tions stimulated by our investigation.

Techniques We extended an existing mean-field model that derives activation ratios

from the probability distributions of synaptic inputs (Gibson and Robinson, 1991; Leibold and Kempter, 2006). We studied how replay evolved with time in mean-field phase space and how it depended on network parameters in the phase diagram from the theory of discrete dynamical systems. Optimal detection theory with the Bayes criterion was used to find the best threshold for excitatory neurons. Cellular simulations were employed throughout as ground truth to check the validity of mean-field findings.

Results Inhibition added to our neural network model for sequence memory indi-

rectly boosted capacity by stabilizing replay modes that were unstable in the excitatory- only network. We showed analytically in the mean field approximation that the best global instantaneous inhibition is given by a threshold that grows linearly with exci- tatory activity. A population of inhibitory neurons that facilitates replay close to the capacity limit displays oscillatory activity, with increasing amplitudes betraying the onset of instability at high loads. Recurrence in the inhibitory population was found to marginally enhance capacity and robustness.

Open questions Some of following open questions exceed the scope of the present

model, as it is difficult to obtain additional understanding of memory networks staying within its assumptions. We wonder how to systematically investigate the effect of the high degree of inhibitory recurrence that seems to be present in biological networks. If replay at high capacity is inherently a transient phenomenon, how are represented extended experiences spanning several replay containers (Davidson et al., 2009)? Is

the frequency relation between excitatory and feedback inhibitory activity dependent on the time-discrete nature of the model? The quantitative parameters of both CA1 and CA3 fit within our general modeling framework, but what so far uncaptured assumption makes biological sequences tend to be expressed rather in CA1?

5.2.2 Replay Benefits From an Adaptive Threshold

Our replay model contains variability derived from the random choice of the patterns and the random morphological connectivity of the neural network. The variability implies that, given a cue assembly, there will be in the target pattern both active neurons that should be inactive and silent ones that should be active. When the number of active neurons required to encode memory is pushed down so as to increase capacity, every neuron is excited close to its threshold: slightly below if it is a nonparticipating neuron and slightly above if it belongs in the target memory. If, due to the randomness, the average activity increases at any given iteration, it will be difficult when the pattern size is ambitiously driven down for extra capacity, to prevent a positive feedback loop from taking over and activating the whole network (conversely for the opposite case leading to a silent network). A threshold proportional to overall activity provides the necessary feedback to keep the network stable at high capacity. But this threshold can only be tuned in a sequence setting, where the previous iteration provides information about the likely level of global activity in the present one. Hence, the temporal continuity of cue-target associations in a sequence enables the choice of a useful threshold, as long as the network is not destabilized in a single step.

Considering each neuron as a binary classifier confronted with a decision task, we employed ROC curves to visualize the need for an adaptive threshold. They showed how an initially near-optimal threshold became progressively unsuitable as replay pro- ceeded.

5.2.3 Static Inhibition and the Optimal Threshold

Instantaneous feedback inhibition (as provided by a hypothetical fast, globally con- nected interneuron) can stabilize the retrieval of memory sequences and thereby increase both memory capacity and robustness. The optimal instantaneous inhibitory feedback is a roughly linear function of the total network activity. This was found out numerically by Hirase and Recce (1996). Here the same conclusion was reached semianalytically using a probabilistic approach in the framework of optimal detection theory. Additionally, we could identify the desirable coupling coefficient to excita- tion as a function of pattern size. Static inhibition admits in a first approximation an interpretation as a reduction of the effective excitatory connectivity, which results in a net increase of the resources for plasticity.

5.2.4 Dynamic Inhibition and Oscillations in the Gamma Range

Extending the model to dynamic inhibition, we find that, at the edges of stable replay, inhibition induces strong oscillations. At a rate of one each 5 to 10 ripple oscillations, they roughly correspond to the gamma range (20-40Hz).

Gamma oscillations are ubiquitous in the brain and their synchronization is gen- erally attributed to local inhibitory networks (Wang and Buzsáki, 1996). Several cog- nitive functions have been related to increased gamma power and coherence, such as sensory integration, attention, and memory (Jutras and Buffalo, 2010). Specif- ically, gamma coherence between subregions in the hippocampal formation and prefrontal cortex has been shown to correlate with involvement in a short-term memory task (Sigurdsson et al., 2010). This finding fits well into the general view of gamma rhythms as a mechanism that facilitates communication between brain areas (Colgin et al., 2009).

In our model, gamma occurs as a side effect of feedback stabilization during replay, and by construction, the rhythm does not entrain the excitatory population. It is a suggestive speculation, based on the reports above, that the enhanced power of a brain rhythm can be a dynamic signature of a functional state, in this case that gamma- frequency oscillations be the signature of memory overloading. If gamma, addition- ally, does indeed correlate with information transfer to other brain areas, then the implication would be that the critically loaded hippocampus periodically rids itself of memories during periods of enhanced gamma (but the hippocampus may still be instrumental for recall, see Goshen et al., 2011). If the establishment of the transfer rhythm is conditional upon overloading at the source, then memory networks would need to be critically loaded for transfer to be initiated and consolidation would be lazy, i.e. activated on an as-needed basis.

5.2.5 Transient Retrieval

Our model identifies parameter regions in which transient retrieval occurs that lasts for only a few time steps. These regions of transient retrieval extend far into low pattern sizes for strong inhibitory feedback, and thus correspond there to the regimes of largest memory capacity. Neuronal networks exhibiting activity sequences hence operate with optimal memory performance in a regime of hyperexcitability that is stabilized by delayed inhibition. This transient retrieval regime is consistent with the dynamic fea- tures of sequence replay during sharp wave ripple complexes in the hippocampus. These last for 5 to 10 cycles of an approximately 200 Hz oscillation. Our data analysis work on an in-vitro model of SWR (see above) shows that they are accompanied by delayed inhibitory feedback, though the frequency is roughly similar to that of excita- tory activation, and the phase detail cannot be compared with a theoretical model that only has one time step per ripple cycle.

A way out of the capacity vs. stability dilemma In large environments, sequence

replay in-vivo can span several ripple episodes (Davidson et al., 2009), allowing long sequences to be constructed by concatenating multiple fragments riding on transient, time-limited ripples. An interpretation of such fragmentation in the context of our theoretical findings is that the dilemma between stability and capacity is solved by chaining several inherently unstable replay epochs, each tuned to maximum capacity. An open question is what is the optimal tradeoff between sequence length and number of successive ripples. At the mechanistical root, it is also unclear how information can be transmitted between successive ripples.

5.2.6 Capacity Bounds Depend on Model Assumptions

Throughout this work, we considered the connectivity parameters c andcm as con-

stants, based on the assumption that the morphological connectivity c_m is mainly determined by geometrical constraints such as the size of the cell surface, or the volume requirement of wiring. A degree c of functional connectivity results from

the specific learning rules that ensure that the network always remains plastic —in order to store new memories a large fraction of synapses has to be able to change its state. In the parameter regime used for our analysis, this requirement is fulfilled by fixing c/cm=0.5. Moreover, the network is sparse, based upon the experi-

mental evidence that indicates that hippocampal networks are sparsely connected (Miles and Wong, 1986).

A specific assumption underlying our analysis is that of a constant and low pattern sizeM. While the firing sparsity of hippocampal principal neurons is well established, remaining even during ripples below 10 Hz (Csicsvari et al., 1999b), the pattern size itself is likely to be variable in reality. This entails a potential loss of capacity.

Another significant simplification is the discreteness in time. Dynamical interac- tions of synaptic currents and membrane processes during sharp-wave ripples may also reduce capacities. In this sense the capacity values that we have derived here can only be considered as upper bounds useful for determining scaling behavior. Extending the model to more realistic dynamics is necessary to investigate how tight is this bound for a real spiking network.

5.2.7 More Sophisticated Models Needed to Formulate Predictions

Our time-discrete model of replay ignores the possibility of desynchronization of assemblies that exists in continuous time, as we have just advanced. But it also limits the conclusions that can be reached about the neurophysiological substrate. First, it makes it impossible to examine the dynamics on time scales shorter than one iter- ation step. In order to be able to accomodate our experimental findings on the phase shift of inhibition with respect to excitation, we need a new model where these fine- grained interactions can be taken into account. Second, not only temporal informa- tion gained from experiment is disregarded, but also the cycle-dependent amplitude of ripple currents.

A framework is needed that can relate the observed currents to coding, i.e. neu- ronal spiking. The mediator between these two objects is, naturally, the neuronal mem- brane potential. In order to estimate it, an integration equation is needed that, given the conductances, can yield the time-dependent membrane potential. At first, a simple equation with few parameters, like the integrate-and-fire model, can be employed. Conductances being notoriously voltage-dependent, it may be necessary to estimate them from a range of holding potentials in voltage clamp experiments, using for instance peeling reconstruction wherever the polarity allows. The resulting integrated membrane potential can be validated by comparison to current clamp experiments.

A probabilistic model based upon ripple-cycle-dependent conductance distribu- tions can tell us how likely is it for neurons to spike at a given cycle in the ripple. The spiking can then be built into a network model that can be explored for self-consis- tency, i.e. the sustained spiking output of the network should lead to the observed transient SWR bursts in the LFP.

Even recent models of ripples, whether based on synaptic, directional (Memmesheimer, 2010; Taxidis et al., 2011) or nonsynaptic, symmetric communica- tion (Vladimirov et al., 2012) do not address the functional aspect of sequence replay. We thus identify an acute need to bridge the observed functional role of SWR (replay of behavioral storylines encoded by spike sequences) and its mechanistic constitution as as short-lived LFP burst of intertwined excitatory and inhibitory activity. The model we suggest would provide a stepping stone towards the incorporation of further bio- logical constraints from other studies of SWR, while keeping the replay capability as a functional constraint. The critical test for the integration of both is the ability of assem- blies to remain well-differentiated and propagate their activities across ripple cycles.