Estimating Memory Deterioration Rates Following Neurodegeneration and Traumatic Brain Injuries in a Hopfield Network Model

Neurodegenerative diseases and traumatic brain injuries (TBI) are among the main causes of cognitive dysfunction in humans. At a neuronal network level, they both extensively exhibit focal axonal swellings (FAS), which in turn, compromise the information encoded in spike trains and lead to potentially severe functional deficits. There are currently no satisfactory quantitative predictors of decline in memory-encoding neuronal networks based on the impact and statistics of FAS. Some of the challenges of this translational approach include our inability to access small scale injuries with non-invasive methods, the overall complexity of neuronal pathologies, and our limited knowledge of how networks process biological signals. The purpose of this computational study is three-fold: (i) to extend Hopfield's model for associative memory to account for the effects of FAS, (ii) to calibrate FAS parameters from biophysical observations of their statistical distribution and size, and (iii) to systematically evaluate deterioration rates for different memory-recall tasks as a function of FAS injury. We calculate deterioration rates for a face-recognition task to account for highly correlated memories and also for a discrimination task of random, uncorrelated memories with a size at the capacity limit of the Hopfield network. While it is expected that the performance of any injured network should decrease with injury, our results link, for the first time, the memory recall ability to observed FAS statistics. This allows for plausible estimates of cognitive decline for different stages of brain disorders within neuronal networks, bridging experimental observations following neurodegeneration and TBI with compromised memory recall. The work lends new insights to help close the gap between theory and experiment on how biological signals are processed in damaged, high-dimensional functional networks, and towards positing new diagnostic tools to measure cognitive deficits.

In Hopfield's original model, a neuronal network is composed of N neurons that attend binary states S i ∈ {−1, 1}. The connections between neurons are responsible for information transference and processing in the network. They are represented by weights w ij (linking neurons i and j), and stored in a connectivity matrix W = (w ij ). In this setting, the neuronal states evolve in time according to (Hopfield, 1982(Hopfield, , 1984Hopfield, Tank, 1985) dS i (t) = j w ij · g(S j (t))dt (S1) where the gain function g is given by The most important property of the model is the ability to encode memories as fixed points of the system.

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When a noisy input is presented, it converges to the closest fixed point (closest known concept) in a process 11 commonly referred as memory association. 12

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Neuronal states are theoretically modeled as continuous spike trains transmitted through axonal channels Adrian (1926); Richmond et al. (1987). In computational studies, these continuous states are discretized for more efficient computability. Hopfield's original model as described in the previous section considers two binary states distinguishing between an 'on' and an 'off' mode. However, the binary model is not rich enough to model more sophisticated injury mechanism, such as filtering and reflexion in the Maia and Kutz theory. While a continuous model was beyond the scope of this study, we implemented a multi-level discrete state model to account for different modes of neuronal activity. In our extended Hopfield model, neurons may achieve multiple discrete states (Gerstner et al., 2014;Benna, Fusi, 2015) S i ∈ {0, 1, ..., s − 1, s}.
The dynamical evolution of the system is also governed by a more sophisticated equation: with sigmoid gain function g given by The constant τ gives the time-scale of the dynamics. Direct inputs for neuron i (e.g. external stimuli) are represented by I i (t). The term B i corresponds to a Wiener Process with intensity µ, and is a proxy for stochastic fluctuations in the firing rates. The (continuous) states are ultimately rounded to the nearest discrete state by a scaling function The resulting stochastic differential equation takes the following form when discretized: 14 We solved the system numerically using the Euler-Maruyama Method (Higham, 2001) and made all our 15 codes available. If higher accuracy is desired beyond the Euler-Maruyama scheme, recent algorithms have 16 been developed to potentially improve accuracy and stability (see (Rößler , 2009) and (Omar et al. , 2011) 17 based on (Milstein , 1975)

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A number of early attempts on modeling associative memory with networks predates key ideas in 31 Hopfield's model (McCulloch, Pitts, 1943;Hebb, 1949;Steinbuch, Piske, 1963;Willshaw, 1969;Little, 32 1974;Kohonen, 1989). Among the very early work is Pitt's 1943 article (McCulloch, Pitts, 1943) that Swellings typically delete spikes by a mechanism called filtering (β 2 ), when a first spike changes its 53 profile at the axonal enlargement region and a close second spike interacts with its refractory period. As 54 a consequence, the second spike is deleted in a mechanism of the so-called pile-up collision (see (Maia,55 Kutz, 2014b) for details). Distorted spike trains do not match their corresponding original firing rates (as 56 illustrated in Fig. S1). Instead, they are confused with lower rates, which decrease the system's overall 57 denoising abilities. We simulate the harmful effects of filtering by implementing a statistical version of 58 the confusion matrix from the same source, that in simple terms, evaluates the probability that state i gets 59 confused as state j due to the FAS.

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A less frequent mechanism of spike deletion is reflection (β 3 ). There, a traveling pulse is divided into . We generate ovoid/spheroid FAS with areas compatible with the experimental distribution. The geometrical parameters of the FAS define the spike propagation regime. Panel C: We generate 12 FAS (column) for each injured axon (row) and order them from worst to best case scenario (upper "flags"). We assume that the worse FAS within an injured axon dominates the others, and classify the entire axon within that category (intermediary "flags"). This leads to the (bottom) pie-charts of impairments for an injured neuronal population. See text for more details.
original encoded information is ultimately transmitted by the spike train. We add this effect in our neuronal 64 network by halving the firing-rate of an injured neuron in this regime.  (Dikranian et al., 2008). We generate ovoid/spheroid FAS following the reported experimental distribution of FAS diameters. The geometrical parameters of the FAS define the spike propagation regime. We generate 5 FAS (column) for each one of the 40 injured axons (row) and order them from worst to best-case scenario (upper "flags"). We assume that the worse FAS within an injured axon dominates the others, and classify the entire axon within that category (intermediary "flags"). This leads to the (bottom) pie-charts of impairments for an injured neuronal population. See text for details. modeled this mechanism by introducing non-adapting neurons into the network, that keep their (possibly 71 noise-affected) initial state over time.

IMPLEMENTATION OF MEMORY STORAGE
To simulate a face recognition task, the set of memories has to be learned by the network. For this, we 77 encode them in the weights of the neuronal connections as specified by the weight matrix of the network: 78 We consider a system of weighted neurons. The strength w ij of the connection between neuron i and neuron The theoretical storage capacity of a (Standard) Hopfield network of size N is 0.14N random patterns. In 83 this study, we use a much smaller set of memories, respectively five and three. This is due to the fact, that 3. Construct weight matrix as 96 W = P T · P .
In our computational experiments, we set N = 900, M = 126 and threshold = 1.5, which yields a 97 sparsity of about 13% in matrix P and a conditionl number of 49.28 .

RECOGNITION SCORE FOR NETWORK PERFORMANCE
We developed a recognition score that measures recognition abilities with respect to significance and 99 accuracy in recalling previously stored memory patterns (see Fig. S3).

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We assume the existence of an ideal observer (cf. Benna and Fusi(Benna, Fusi, 2015)), that knows the  Figure S3. Calculation of recognition scores for measuring memory performance (see Hopfield Recognition Toolbox, current version available at GitHub: https://github.com/MelWe/ hopf-recognition). We use the Hamming distance m µ i to measure the overlap between the current network state and the fixed points corresponding to known facial images. Confusion or recognition is characterized by m µ i : if the overlap with the correct facial image is highest, we speak of recognition, otherwise of confusion. A threshold for the difference between the highest and second highest overlap determines whether the recognition or confusion was significant. According to this classification, we assign color labels to each trial which can be displayed in a heat map.
(ii) Recognition and Significance: After a pre-defined number of time steps (system's parameter), the network's states are matched to the closest pattern, i.e., we determine the µ ∈ 1, ..., M , such that d µ = |m orig − m µ | is minimal.
If the output pattern matches the original one (µ ≡ orig), we say that recognition occurs. Otherwise, 105 we speak of confusion of the memories (concepts). The classification is considered significant only if 106 |d µ − d i | < t ∀i = 1, ..., M ; (µ = i), where t is a threshold parameter. With this scheme, we classify the memory recall into four groups and 107 assign (numerical) labels.

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(iii) Evaluation: The recognition score was developed to evaluate the memory performance of our Hopfield 109 neuronal network model over a broad range of injury (parameter inj) and initial noise (parameter noise).

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For each pair of parameters (inj, noise) we calculate the score as value of the significance label scaled 111 by the accuracy of the recognition (overlap m µ ).

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The final result is a heat map (see Fig. 2,3 in main text) that links recognition score, memory performance 113 and noise handling to different levels of injury. p.442-66.