- Research article
- Open Access
Imbalanced pattern completion vs. separation in cognitive disease: network simulations of synaptic pathologies predict a personalized therapeutics strategy
BMC Neurosciencevolume 11, Article number: 96 (2010)
Diverse Mouse genetic models of neurodevelopmental, neuropsychiatric, and neurodegenerative causes of impaired cognition exhibit at least four convergent points of synaptic malfunction: 1) Strength of long-term potentiation (LTP), 2) Strength of long-term depression (LTD), 3) Relative inhibition levels (Inhibition), and 4) Excitatory connectivity levels (Connectivity).
To test the hypothesis that pathological increases or decreases in these synaptic properties could underlie imbalances at the level of basic neural network function, we explored each type of malfunction in a simulation of autoassociative memory. These network simulations revealed that one impact of impairments or excesses in each of these synaptic properties is to shift the trade-off between pattern separation and pattern completion performance during memory storage and recall. Each type of synaptic pathology either pushed the network balance towards intolerable error in pattern separation or intolerable error in pattern completion. Imbalances caused by pathological impairments or excesses in LTP, LTD, inhibition, or connectivity, could all be exacerbated, or rescued, by the simultaneous modulation of any of the other three synaptic properties.
Because appropriate modulation of any of the synaptic properties could help re-balance network function, regardless of the origins of the imbalance, we propose a new strategy of personalized cognitive therapeutics guided by assay of pattern completion vs. pattern separation function. Simulated examples and testable predictions of this theorized approach to cognitive therapeutics are presented.
Impaired cognition occurs in many different neurodevelopmental, neuropsychiatric, and neurodegenerative diseases. The identification of numerous disease-linked gene mutations has led to the creation of various transgenic mouse models that replicate the phenotypes of human patients, especially learning and memory impairments. Our analysis begins with a review of existing neurophysiological and neuroanatomical experiments in diverse genetic models of impaired cognition that include memory deficits. This synthesis of the literature highlights four properties of synaptic or neural network function that are commonly altered in these conditions: 1) Strength of LTP, 2) Strength of LTD, 3) Relative inhibition, and 4) Connectivity levels (Table1).
To explore potential network level impacts of these four convergent points of synaptic pathology, we examined the performance of a neural network simulation of autoassociative memory while varying the strength of each synaptic property. Associative memory requires binding separate elements of a sensory experience into a single memory that can be later recalled in its entirety, even when cued by only some of the original elements. Autoassociation is the ability of neural networks to perform associative memory without any external guidance, via changes in synaptic strengths caused by neuronal activity. Autoassociative memory in area CA3 of the hippocampus in particular, is perhaps the best example of a convergence of theoretical predictions [1–3] and experimental evidence [4–8] for how neural networks store memories, and is thought to represent a basic process essential to the learning capabilities of interconnected brain networks [9, 10]. Two key features supported by autoassociative memory are pattern separation, and pattern completion. Pattern separation is the ability to keep distinct memory patterns separate during storage (Fig.1A,B), while pattern completion is the ability to recall an entire stored memory pattern in response to a degraded or partial observation of elements of the stored pattern (Fig.1C,D). Analytical models of autoassociation have described a trade-off between pattern completion and separation functions that is influenced by the strength of LTP and LTD . Previous work also indicates that autoassociative network capacity is dependent on both excitatory connectivity levels and the properties of synaptic inhibition [12, 13]. Therefore, we hypothesize that the cognitive diseased-linked synaptic pathologies of LTP, LTD, inhibition, and connectivity should all converge in affecting memory performance by shifting the underlying trade-off between pattern completion and pattern separation. In particular we predict that while the net effect of some constellations of pathologies will be severe deficits in pattern separation, other constellations of pathologies will impair memory performance due to severely impaired pattern completion. Moreover, regardless of the underlying deficits, appropriate manipulation of any of the four synaptic functions could help correct imbalanced autoassociative function. These hypotheses of pathology and therapeutics were tested using network simulations based on current concepts of autoassociative function.
Diverse disease models exhibit convergent synaptic and circuit alterations
Examples of neurodevelopmental diseases that can include memory deficits, where causative genes have been identified and mouse models have been created, include Angelman syndrome, Down syndrome, Fragile × syndrome, FRAXE Syndrome, Rett Syndrome, Neurofibromatosis, Tuberous Sclerosis, and various X-linked Mental Retardations (XLMR) (Table 1, top). Transgenic mouse models have also been created that are relevant to neuropsychiatric conditions including schizophrenia, a disease where memory impairment is an important endophenotype (Table 1, middle). In addition to Alzheimer's disease, other neurodegenerative diseases often more noted for their hallmark motor symptoms, also feature important cognitive phenotypes, and mouse models of neurodegenerative conditions with memory alterations include Amyotrophic Lateral Sclerosis (ALS), Huntington's disease, Parkinson's disease, and Spinocerebellar Ataxia (SCA) (Table 1, bottom). Together, these diverse mouse models provide a comparative window into potential substrates of memory impairment. In particular reoccurring points of pathological changes include: 1) Strength of LTP, 2) Strength of LTD, 3) Relative inhibition, and 4) Connectivity levels.
Neural Network Simulation of Autoassociation
To explore the impact of synaptic alterations on autoassociative functions, we used a neurobiologically realistic, reduced network model of autoassociation that allowed modulation of each of the four points of synaptic pathology. This model was based on a published model of hippocampal area CA3, in which the subpopulations of neurons that are active during distinct gamma cycles are the substrates of memory storage and recall [13, 14]. While replicating the concept of this previous model, our implementation used simplified biophysically realistic neurons with properties taken from a model of rhythm generation in the hippocampus  (see methods, Fig.2). It should be noted that the model used here resembles previous models that use biophysically detailed individual neurons within a simplified neural circuit [16, 17], rather than models with more complex network interactions and more sophisticated memory function, but simpler single neuron representations [18–20]. All of our simulations focused on networks of 100 excitatory neurons with feedback inhibition, and each pattern to be stored and recalled consisted of the activation of a set of 10 individual neurons (Fig.2,3). Each of the four synaptic phenotypes observed to be increased or decreased in mouse models of cognitive impairment corresponded directly to variables within the simulations as follows:
The connectivity level of the recurrent excitatory synapses was set to a given value (for example 50%; Fig.3A), which was implemented by constraining the average number of randomly selected postsynaptic target neurons for each presynaptic neuron.
LTP and LTD
In the absence of any stored memories, all potential synaptic connections were silent with no AMPA receptor (AMPAR) conductance. LTP was implemented by potentiating synapses between presynaptic and postsynaptic neurons that both fired action potentials within the time window of a single gamma cycle during pattern storage. The strength of LTP was defined by the maximal excitatory synaptic conductance variable, gMaxAMPA. LTD was implemented by depressing synapses between presynaptic neurons and postsynaptic neurons that were active during storage of different stimulus patterns within a set of simultaneously stored patterns. The parameter, γLTD, defined the strength of LTD (see methods, Fig3B). The result of these plasticity mechanisms is that as larger numbers of patterns are simultaneously stored in the network, more silent synapses are potentiated, and more of those potentiated synapses are also reduced in strength by LTD due to accumulated overlap of the stimulus patterns (Fig.3C,D).
The strength of the feedback inhibitory conductance, gGABA, received by each pyramidal neuron was set relative to the average maximal excitatory conductance received by excitatory neurons, as defined by the Inhibition Ratio variable (see methods). Thus, the strength of LTP, LTD, connectivity, and inhibition could all be varied to simulate pathological conditions or therapeutic modulation via manipulation of single variables.
Measuring Pattern Completion and Pattern Separation
Sensory stimulus patterns representing distinct memories each consisted of the direct activation of 10 of the 100 neurons in the network. Sets of stimulus patterns, equivalent to a list of multiple memories to be maintained simultaneously, were selected at random with larger sets having greater average overlap between the stimulus patterns. Sets of patterns were stored in the network by calculating and applying the synaptic plasticity resulting from the storage of an entire set of stimulus patterns (for example a set of 5 patterns each consisting of the activation of 10 neurons). Since interleaved learning was assumed, the dynamics of sequential storage of different patterns were not modeled . Pattern separation was tested by activating the 10 neurons of each stored pattern, and evaluating any spurious firing in the remainder of the network during the time window corresponding to the peak of a single gamma cycle (Fig.4A). Activation of one or more neurons not participating in a pattern was considered separation failure for that pattern. Pattern completion was evaluated by activating 9 out of 10 neurons in a stored pattern, and checking for evoked firing of the 10th neuron of that pattern within the same gamma cycle (Fig.4B). Lack of firing of the 10th neuron during that gamma cycle was considered pattern completion failure. This reduced network of 100 neurons using stimuli consisting of activation 10 neurons was sufficiently complex to model pattern completion and separation analogous to that depicted in Fig.1. Rates of pattern completion and pattern separation errors per memory pattern as a function of the number of simultaneously stored stimulus patterns were measured for each set of simulation conditions (see methods, Fig.4C,D). Initial simulations revealed that both pattern completion and pattern separation error rates increased as the number of simultaneously stored memories increased.
Wild-type Synaptic Properties
While biologically plausible values of each synaptic property serve as a good starting point for the baseline model, the necessary simplifications of the model make it difficult to predict exact wild-type values of synaptic properties. For example, relative inhibition is specified using the ratio of GABAA receptor (GABAAR) to AMPAR conductance in each spatially reduced neuron. However the dynamics of synaptic interaction in spatially complex neurons, can enhance the ability of inhibition to oppose excitation [21, 22], suggesting amplified GABAAR to AMPAR conductance ratios may be needed in the simplified simulation. Because of such considerations, we decided to determine wild-type network parameters via an empirical assessment of storage and recall performance. To avoid focusing on a non-unique set of optimal parameters we explored network performance over a broad range of parameter space by creating a database with 960,000 combinations of parameter values (gMaxAMPA, γLTD, Relative Inhibition, and Connectivity %) and memory storage conditions (see methods). While different metrics could be used to assess network performance, we defined optimal networks based on the maximal error rate, which was calculated as the greater of pattern completion or pattern separation error rates. Using this measure, pattern completion and pattern separation errors were equal in their ability to limit network performance.
In the initial evaluation of optimal parameter combinations, 100% connectivity levels were always found to allow the best performing networks, although this required extremely low values of LTP, and extremely high values of LTD and inhibition. However, anatomically-based estimates of in vivo connectivity within CA3 are more sparse, with estimates ranging from very low up to 50% [13, 23].The higher value of 50% connectivity is consistent with physiological observations of ~50% in cultured brain slices that have re-grown CA3 connections severed during slice preparation, but maintain the total synaptic input levels seen in acutely prepared slices [24, 25]. Supporting an upper limit on total connectivity, pathological increases in connectivity within neuronal subpopulations are often accompanied by decreased connectivity in other subpopulations [24–26]. Therefore, to maintain biological realism and minimize neuronal number in the simulations, while defining wild-type networks, we searched the parameter space of the other three synaptic properties, while constraining wild-type connectivity levels to 50%. Optimal balanced networks were found by searching for parameter combinations that produced the lowest maximal error (the higher error in either pattern separation or pattern completion). This analysis revealed that a relatively broad contour of LTP, LTD, and inhibition, parameter space could support optimal balanced network function (Fig.5A). Thus, the illustrated 100 best parameter combinations reflect a diversity of approximately equally well-balanced wild-type networks, perhaps analogous to variability in the in vivo networks of healthy individuals (Fig.5B). Accordingly, in assessing the impact of simulated synaptic pathologies, we considered the average effect of pathologies to each of these 100 optimal 'wild-type' parameter combinations.
Pattern Separation and Completion with Single Synaptic Pathologies
To test the isolated impact of each synaptic pathology on memory performance, we analyzed pattern completion and separation error rates as LTP, LTD, inhibition, and connectivity, were each varied relative to the optimal wild-type networks (Fig.6). These simulations revealed a striking trade-off between pattern completion errors and pattern separation errors. In particular, increasing or decreasing the value of any given synaptic function could decrease one type of error, but at the expense of increasing the other. High connectivity, strong LTP, weak LTD, or weak inhibition, all reduced pattern completion errors, but did so at the expense of increased errors in pattern separation. Conversely, low connectivity, weak LTP, strong LTD, and strong inhibition could all reduce pattern separation errors, but at the expense of increased errors in pattern completion (Fig.6). The observed extreme error rates in either pattern completion or separation, resulting from the synaptic pathologies are predicted to be sufficient to impair performance on standard memory tasks. On one hand, low rates of separation errors during storage are irrelevant in the face of intolerable rates of completion failure during recall. Conversely, low rates of completion error during recall would be masked by intolerable rates of separation failure during storage. Thus, imbalanced networks with a bias towards either pattern separation or pattern completion are potential substrates of the learning and memory impairments observed in neurological disease.
Interaction Between Multiple Synaptic Alterations
Because the synaptic pathologies all shifted the trade-off between completion and separation, we examined the interaction of multiple simultaneous pathologies (Fig. 7). This analysis of simultaneous increases or decreases in all combinations of the four synaptic properties showed that any two synaptic alterations that could both alone caused a separation or completion bias, would together cause an exacerbation of the bias. Conversely any two synaptic alterations that caused opposites shifts in network balance had the potential to at least partially offset each other. This finding of interchangeable effects of simultaneous synaptic alterations has implications for compensatory mechanisms and therapeutic re-balancing of network function. Given that intolerable levels of error in either pattern separation or pattern completion could be limiting factors in memory performance, manipulations that shift the balance so as to ameliorate extreme imbalances should be of therapeutic benefit. Specific examples of imbalances causing increased maximal error resulting from various synaptic pathologies, along with therapeutic ameliorations of these effects by manipulation of independent synaptic properties are shown in Fig.7. In such attempts to correct networks disrupted by one abnormal synaptic property via manipulations of independent synaptic properties, more balanced minimization of both types of error can be achieved, even if some residual increased errors relative to the optimal networks remain.
Discussion and Conclusions
Substrates of Imbalanced Network Performance
The simulation results demonstrate that increases or decreases in LTP, LTD, inhibition, or connectivity, as observed in mouse models of disease-linked mutations, can shift the balance of autoassociative function towards intolerable error in either pattern completion or pattern separation. However, in individual patients, the impacts of disease-linked mutations will likely be modulated by multiple genetic and environmental factors. For example, even when identical mutations are examined in different background mouse strains (different genetic contexts) opposite pathologies in connectivity are observed within models of Fragile × Syndrome [27, 28] or AD . In addition, many synaptic phenotypes develop with age, indicating that disease-linked mutations have different impacts during development and aging, perhaps in part due to induction of compensatory mechanisms . The various factors that could contribute to manifestation of pathology in the four synaptic phenotypes are illustrated in Fig.8A. Regardless of the underlying origins, however, the ability of multiple synaptic alterations to compound or counteract each other in shifting the balance between pattern completion vs. separation errors (Fig.7) suggests that a large number of combinations of synaptic pathologies will have a net effect of a bias towards either pattern completion or pattern separation.
To intuitively understand the basis of the interactions between the different points of synaptic pathology it is important to appreciate that the underlying substrates of successful pattern completion (and failed separation) necessarily converge at action potential generation, while essential to pattern separation (and failed completion) is lack of inappropriate spiking. Thus while the effects of LTP, LTD, inhibition and connectivity levels all impinge on the ability of synaptic inputs to drive output spiking, other perturbations that also effect input-output coupling will also alter the balance between completion and separation. For example, while beyond the scope of the present analysis, neuromodulatory influences that alter intrinsic excitability are also aberrant in disease states and are expected to alter pattern separation and completion functions [19, 31].
Theory of Personalized Therapeutics
That the four examined points of synaptic pathology all further converge in causing one of two distinct network imbalances provides a potential point of therapeutic intervention (Fig.8B). In particular, a pattern completion bias predicts one direction of therapeutic manipulation for each synaptic property, while a separation bias predicts therapeutic value for the opposite directions of manipulation (Fig.8C). If pattern completion vs. pattern separation performance were to be extensively evaluated in cognitive disease, two general possibilities exist: 1) Some causes of disease will consistently involve a separation bias, while other causes will always involve a completion bias, 2) Even within the same disease, the complex interaction of genetics and environment with disease progression will result in some patients with a completion bias and others with a separation bias.
In the case of uniform biases within a disease this model is useful in predicting cross-therapeutic value of drugs with different targets. All of the reported synaptic phenotypes in AD would push network balance towards a separation bias predicting unambiguous therapeutic targets (Fig.8C, top), which is supported by the observation that decreasing inhibition rescues memory performance in mice modeling AD . However, in other disease models, phenotypes causing opposite biases co-exist (Table1), and the appropriate therapeutic targets are thus unclear because they will depend on the relative magnitude of the different pathologies. For example Down syndrome model mice exhibit AD-like phenotypes of reduced LTP, enhanced LTD, and enhanced inhibition, but also exhibit enhanced recurrent connectivity that would favor a completion bias. Insight comes from the observation that as with AD model mice, GABAAR antagonists can rescue impaired memory in the Down syndrome model mice [33, 34] which is consistent with a net pattern separation bias. Therefore, in addition to drugs decreasing inhibition, the theory predicts that drugs enhancing LTP, and connectivity, and/or reducing LTD should also be of therapeutic value in Down syndrome (Fig.8C, top; Table2). In another example, schizophrenia models exhibit phenotypes both supporting a separation bias (impaired LTP, and decreased connectivity) and supporting a completion bias (impaired LTD, and decreased inhibition). In this case, that enhancement of GABAAR function can rescue impaired memory in Schizophrenia patients[35, 36], suggests a net completion bias dominates. Therefore, along with positive modulators of GABAARs, drugs that would selectively reduce LTP, enhance LTD, and/or decrease connectivity are also predicted to be valuable for treating cognitive deficits in Schizophrenia (Fig.8C, bottom; Table2). Further examples of extrapolating cross-therapeutic efficacy come from mouse models that have been treated with manipulations that can increase LTP. In particular, enhancements of LTP by PAK inhibition in Fragile × Syndrome model mice , by BDNF increases in Huntington's disease model mice [38, 39], or by reduction of αCaMKII inhibitory phosphorylation in Angelman syndrome model mice , are all accompanied by rescued memory performance. These findings are consistent with an underlying pattern separation bias in these conditions, and predict efficacy of the corresponding list of therapeutic targets (Fig.8C, top; Table2).
In the case of non-uniform biases within a disease group, assaying pattern completion vs. separation would be especially valuable for prescribing personalized therapeutics. Moreover, this approach could allow therapeutic prescription even in patients with learning and memory impairments with unknown etiologies and no clear disease diagnosis. Fig.9 illustrates this personalized therapeutics approach using the example of a heterogeneous group of patients with the type of pathologies seen in Schizophrenia models (Table1). Genetic and environmental diversity is represented by starting with the 100 optimal networks with different underlying combinations of synaptic properties, and by implementing varying degrees of the pathological decreases in each synaptic pathology as observed in mouse models of Schizophrenia. While the average autoassociative function in the simulated Schizophrenia cohort reflects a completion bias, high variability between the individuals is evident (Fig.9B). Simply treating the entire group with manipulations such as enhanced inhibition, would results in some therapeutic rebalancing of the group on average, consistent with the findings that GABAAR positive modulators can improve memory performance in schizophrenia patients. However, assay of individual autoassociative biases in each simulated patient leads to the identification of both patients with completion biases that would especially benefit from manipulations including increased inhibition, as well as a some patients with separation biases who would benefit from opposite manipulations including decreased inhibition (Fig. 9C,D).
How pattern separation and pattern completion deficits in neural networks will read out in indirect behavioral measurements such as tests of prospective interference may be complicated, but could be determined by experiments correlating behavior and read-outs of neural network separation/completion function. However, the promise for implementing such an approach of assay-based therapeutic prescription is good, since non-invasive touchscreen-based memory tests already exist, including explicit measurement of pattern separation function, in both mouse models  and human patients [42, 43]. While higher-order processing strategies could confound behavioral read-outs of basic network functions , direct assay of autoassociative function can be performed in rodent models using recordings of neuronal ensemble activity [4–6], and has been demonstrated in humans using functional imaging . Although the current simulations focused on autoassociative function in the key region of CA3, interactions across multiple neural circuits are known to underlie cognitive behaviors like learning and memory. Nonetheless, much of the cortex is organized in recurrent circuits, and could process and store information in a sparser but analogous manner to CA3. Therefore, especially in cases contributed to by genetic disruptions with potentially widespread effects, the predicted manipulations aimed at rebalancing function could be broadly beneficial across neural networks underlying a pathologically extreme cognitive style. Ultimately tests of the types of predictions outlined in Fig.8 and detailed above, will support or refute this theorized approach of personalized cognitive therapeutics.
Network simulations were constructed using NeuroConstruct  and simulations were run using Neuron . Custom Matlab (Mathworks) scripts were used to generate stimulus pattern sets, calculate synaptic plasticity, and analyze simulation output. Each neuron was modeled as an isopotential sphere with a radius of 10 μm and had a membrane capacitance of 1.0 μF/cm2 and contained a leak conductance with Eleak = -67 mV (GLeak = 0.1 μS/cm2), and Fast Na+ (GNa = 100 μS/cm2), and Delayed Rectifier K+ (GK = 80 μS/cm2) conductances, based on a reduced model of hippocampal rhythm generation  (see Fig.2). Na+ current was calculated as: INa = GNam3h(Vm-ENa), with ENa = 90 mV, K+ current was calculated as: IK = GKn4(Vm-EK), with EK = -100. AMPA conductances of excitatory synapses had time courses described by GAMPA = exp(-t/tau2) - exp(-t/tau1), with tau1 = 1 and tau2 = 4, EAMPA = 0 mV, and GABAA conductances had time courses described by: GGABA = exp(-t/tau2)-exp(-t/tau1), with tau1 = 2 and tau2 = 8, EGABA = -80 mV. Synaptic delays were 1 ms and axonal conduction times were considered negligible.
For each simulation of a given connectivity level, three different connectivity profiles were generated using different random seeds in NeuroConstruct. The average performance of simulations using 10 different random sets of memories in each connectivity profile was calculated. Error rates are presented as mean ± SEM of the average performance in three connectivity profiles.
The weight of associational connections (W) following synaptic plasticity was determined using an adaptation of the equation used in the model of autoassociation we based our simulations on : Wij = nij11/(nij11*γ11+nij10*γ10+nij01* γ01), where nij11 is the number of patterns where presynaptic(i) and postsynaptic(j) neurons fire together, and nij10 and nij01 are the number of patterns where presynaptic or postsynaptic neurons fire independently within a set of stored patterns. For simplicity in our implementation, γ11 was set to a value of 1, and γ10 and γ01 shared the same value, γLTD. This yielded the equation; Wij= gMaxAMPA *nij11/(nij11+(nij10+nij01)*γLTD), where the normalized strength of maximal potentiation, gMaxAMPA, defines the strength of LTP, and the single parameter, γLTD, defines the strength of LTD (Fig.3B).
The strength of the excitatory synapses onto the inhibitory neuron was set to 90% of gMaxAMPA. The strength of inhibitory synaptic conductances in the network was set relative to the maximal total excitation received by a pyramidal neuron during a stimulus pattern involving 10 neurons such that: gGABA = gMaxAMPA*10*Connectivity Level*Relative Inhibition.
Parameter permutations consisting of 20 connectivity levels spaced between 5 and 100%, 20 gMaxAMPA values spaced between 2.78 and 55.56 nS, 20 logarithmically spaced Inhibition Ratio values between 0.01 and 100 (corresponding to 1.83 fold increments), and γLTD values logarithmically spaced between 0.1 and 10 (corresponding to 1.27 fold increments) were evaluated. For each of these 160,000 permutations of synaptic properties, 6 different sized pattern sets were assessed using multiple connectivity and stimulus pattern random seed conditions (see connectivity), for a total of 180 simulations per parameter combination. To facilitate the required large number of simulations, an approximation was made, based on the fact that the presence or absence of a spike in each individual neuron (which determines pattern storage or pattern completion success or failure), is determined entirely by, 1) the total excitatory conductance resulting from the properties of synaptic plasticity in the context of connectivity, and 2) the strength of inhibition. Therefore, a table of spike thresholds as a function of both total excitatory and inhibitory conductance received by a neuron was generated from a set of simulations with systematic variations in these properties. During creation of the database, pattern separation and pattern completion errors were assessed based on neuronal firing patterns determined by comparing values of total excitatory and inhibitory conductances in each neuron to the table of firing thresholds. Validation of this efficiency measure was performed by directly comparing several key parameter combinations using the threshold-table approximation with full explicit simulations. Optimal balanced networks were defined as the parameter combinations with the lowest maximal error (the higher error in either pattern separation or pattern completion), thus reflecting an even breakdown of pattern separation and completion for a given number of stored patterns.
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This work was supported by grants from the National Institute of Mental Health (MH065541) and by The G. Harold and Leila Y. Mathers Charitable Foundation.
The authors declare that they have no competing interests.
JEH conceived of the study, participated in the design of the study, carried out the computational simulations, participated in the interpretations of the results and drafted the manuscript. DVM participated in the design of the study and interpretation of the results and contributed to the intellectual content of the manuscript. All authors read and approved the final manuscript.
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