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Edited by: Clive R. Bramham, University of Bergen, Norway

Reviewed by: Panayiota Poirazi, Foundation for Research and Technology Hellas, Greece; James B. Aimone, Sandia National Laboratories (SNL), United States

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Structural plasticity, characterized by the formation and elimination of synapses, plays a big role in learning and long-term memory formation in the brain. The majority of the synapses in the neocortex occur between the axonal boutons and dendritic spines. Therefore, understanding the dynamics of the dendritic spine growth and elimination can provide key insights to the mechanisms of structural plasticity. In addition to learning and memory formation, the connectivity of neural networks affects cognition, perception, and behavior. Unsurprisingly, psychiatric and neurological disorders such as schizophrenia and autism are accompanied by pathological alterations in spine morphology and synapse numbers. Hence, it is vital to develop a model to understand the mechanisms governing dendritic spine dynamics throughout the lifetime. Here, we applied the density dependent Ricker population model to investigate the feasibility of ecological population concepts and mathematical foundations in spine dynamics. The model includes “immigration,” which is based on the filopodia type transient spines, and we show how this effect can potentially stabilize the spine population theoretically. For the long-term dynamics we employed a time dependent carrying capacity based on the brain's metabolic energy allocation. The results show that the mathematical model can explain the spine density fluctuations in the short-term and also account for the long term trends in the developing brain during synaptogenesis and pruning.

Rewiring of biological neural networks via structural plasticity is a fundamental mechanism for continuous learning (Bremner,

The well-known rules of learning such as the Hebbian learning and spike timing dependent plasticity (STDP) are based on pre- and post-synaptic activity. Synaptic properties can also change spontaneously (i.e., regardless of activity), reinforcing the “dynamic synapse” (Choquet and Triller,

Two different dendritic protrusions in contact with nearby axons. Left is a motile filopodium, which is exploring an axonal partner to form a synapse. On the right, a mushroom shape spine is illustrated, which made a synaptic contact to an axon.

In this work, we will focus on the dynamics of dendritic spine populations and develop a mathematical model based on some of the well-known population ecology concepts. The model will cover both the short and the long-term dynamics. By “short-term,” we refer to the local dendritic spine density changes that occur within hours or days, typically in response to a sensory experience or learning task. On the other hand, the global spine density changes that happen in years is referred as “long-term,” which defines the lifetime trajectory of synaptic connectivity.

The dynamics of dendritic spines have been previously studied by Yasumatsu et al. (

Some other computational approaches (Verzi et al.,

A more general rule for spine population changes and cortical rewiring has been proposed by Butz and van Ooyen (Butz et al.,

Contrary to the prior approaches, in this work, we will ignore the stochastic fluctuations of spine volume and focus on the number of spines on a dendritic branch driven by new spine growth and spine elimination. Moreover, we will present a deterministic model, which can be used to predict the spine density trajectory over a long time (potentially the lifetime). Homeostasis is a crucial force in shaping the population in many species, therefore, the rule proposed by Butz and van Ooyen can potentially be incorporated in our model.

The population dynamics models have mature mathematical foundations and successful applications in a diverse set of practical problems even outside of biology (Neal,

A population, which can be defined as a collection of individuals of a particular species living in a well-defined area, is subject to four factors that can change its size and density. This change can formally be expressed with the following balance equation:

We will consider dendritic branches as “well-defined areas” for spine populations, which are typically the subject of experimental studies looking at spine densities. We will model new spine growth as “birth” and spine elimination (pruning) as “death” (Figure

This diagram illustrates the structural changes observed on a dendritic branch during spine generation and elimination.

The concepts of “immigration” and “emigration” in population biology may seem challenging to apply in the dynamics of dentritic spines. In the model we are suggesting, fast appearance and disappearance of filopodia could be viewed as immigration and emigration, respectively (Figure

Next we need to consider the other factors for the density dependence of the population growth. The function of synaptic connections on a dendrite is communication with other neurons. Based on the activity dependent learning rules, such as Hebbian and spike timing-dependent plasticity (STDP) (Caporale and Dan,

Populations cannot grow forever and limited resources (e.g., available food) usually limit the growth of the population. In the adult brain, the overall synapse (as well as spine) density is relatively stable over long time scales under normal conditions (Petanjek et al.,

In the light of these considerations, we decided to start with the Ricker population model (Ricker,

where _{t} is the population size at time

One of the interesting questions for population dynamics is stability. The population size in the Ricker model is stable for low

Ricker maps for Equation (1).

Chaotic oscillations of the population density may result in overpopulation or extinction. In the context of spine populations, abnormal levels of spines are directly linked to neuropsychiatric diseases such as autism, schizophrenia and Alzheimer's disease (Penzes et al.,

For a population that cycles in time and even displays erratic chaotic behavior, dampening the oscillations is crucial for stability. A number of studies (Ruxton and Rohani,

In order to introduce immigration in the Ricker model we can modify Equation (2) by adding a constant parameter,

Plotting the population change as a function of time for

Ricker maps to illustrate the effect of immigration.

A key part of our hypothesis is based upon the observation that there are “activity dependent” and “spontaneous” (i.e., activity independent) spine growth. Similarly, others have referred to “learning spines” and “memory spines” which correspond to thin and mushroom shape spine morphologies respectively (Bourne and Harris,

So far we have focused on the short-term and local dynamics of spine populations, which is typically the subject of

The statistics of the data in most of the above mentioned studies are not sufficient to estimate the functional form of the spine density evolution. To our knowledge, only Petanjek et al. (

The generally accepted view of the spine density evolution includes a sharp increase in the spinogenesis rate during the early years and then a pruning period (Bianchi et al.,

In the Ricker growth equation we described above (Equations 2, 3), the carrying density was assumed to be constant. However, it is reasonable to consider a dynamic time-dependent carrying capacity for the spine population that depends on time. The most obvious parameter that we can relate to spine carrying capacity would be the available metabolic energy. Most of the energy consumed by the brain is due to spiking of neurons for communication (Attwell and Laughlin,

Previous estimates of the percentage of resting metabolic rate (RMR) allocated to the brain per weight suggested a constant decreasing rate from infancy to adulthood (Holliday,

The percentage of resting metabolic rate (RMR) allocated to the brain as a function of age (data from Holliday, ^{b}. The fitting gives

Based on the fit from the RMR percentage by age, we used the same exponent (i.e., −0.2) for the carrying capacity function in our model, which decays by a power law as a function of time. To highlight the time dependence of K, Equation (1) will be modified as:

where, _{0} is the initial carrying capacity and

Using Equation (4) (no immigration), we have simulated the population change for different growth rates, r, to show how the time dependent carrying capacity affects population growth (Figure

Population change as a function of time without immigration (Equation 4) for various growth rates (

Next, we have included the immigration effect to see how it would change the population dynamics in the presence of a time-dependent carrying capacity. Biological data show that filopodia type protrusions are abundant in the post-natal period but decreases quickly during the brain development. For example studies on mice by Zuo et al. (

where,_{0} is the initial filopodia density and _{0} is the initial carrying capacity and

Figure

Population change as a function of time with immigration (filopodia) (Equation 5) for different set of parameters.

Since our choice for the carrying capacity exponent is only speculative, we explored several values of a to see the impact. Figure

Next, we obtained the published dendritic spine density data from Petanjek et al. (

Spine density data obtained from Petanjek et al. (

Spine density data obtained from Huttenlocher (

In the mammalian brain the majority of the excitatory synapses occur on dendritic spines (Yuste and Bonhoeffer,

In order to model the complex spine dynamics, we have proposed to use the framework of population ecology. There are many factors to consider before choosing and building the right model. First of all, we made the assumption that the dendritic spine population will be density dependent. We justify this based on two observations: 1-Spine density affects connectivity of the neuron and neural activity is the primary factor for the growth of new spines. 2-

Based on these assumptions and observations, we have chosen the Ricker model to apply for modeling dendritic spine population dynamics. Ricker's model is based on a density-dependent discrete equation, which was originally developed for predicting recruitment to fishery stocks (Ricker,

Even though neurons have thousands of synapses, we know that the number of spines saturate at an optimal level, which may depend on the type, age and the location of the neuron. Therefore, it is natural to consider a regulating factor. The Ricker model includes a carrying capacity parameter, which defines the maximum population level that can be sustained. For the short-term spine dynamics we assumed a constant carrying capacity number, which is typical in most applications of the Ricker model.

It is well known that Ricker's equation is stable for smaller growth rates but the population density starts to oscillate at higher rates, which becomes chaotic after a threshold. If the population is not closed, the influx of a small number of immigrants can stabilize the oscillations and prevent chaos, which was observed in island populations and has been studied in detail mathematically (Stone and Hart,

We first showed the results of our calculations without immigration, which is basically the standard Ricker model. For the long-term dynamics, the carrying capacity is modeled as a decaying power function. We base this choice on the estimates for brain's energy demand per weight and its expected trajectory from infancy to adulthood. Since this decay is very slow, the carrying capacity is assumed to be constant at a certain age when we discuss the short-term dynamics. We would like to stress that the constant decay is the trend we use for the carrying capacity change by age and not the actual energy consumption. The actual energy use, which can be estimated based on the glucose uptake, will be related to the spine density and follow a different trajectory (i.e., the trajectory of the population), as confirmed by the recent work of Kuzawa et al. (

Our results with the immigration factor (i.e., the filopodia) is intriguing given the observed impact on the lifetime spine population trajectory. For example, when the filopodia percentage declines faster during the spinogenesis period, the pruning trend is steeper and shifts the population curve, even though the growth rate and the carrying capacity were kept the same. This could potentially explain how spine density abnormalities can occur and cause psychiatric disorders. The root cause for the difference in the filopodia exponent could be genetic but nevertheless, the model provides key insight on the mechanisms that control the spine density. The observed prolonged pruning and maturation of the human prefrontal cortex can also be reproduced by this model by adjusting the filopodia parameter. Dumitriu et al. (

The ideas presented here are clearly speculative and need to be tested with experimental studies where filopodia and spine dynamics are monitored carefully both at short and long time scales. However, the simplicity of the model which is based only on several parameters, is attractive to consider. If our intuitions regarding the filopodia dynamics and the carrying capacity are correct, controlling the filopodia density and metabolic energy resources can be crucial for treating or preventing neuropsychiatric disorders.

Experimental evidence shows that synaptic events are highly dynamic from local alterations to larger topographical changes. One of the key mechanisms that underlies these changes is dendritic spine activity. We have applied a variation of the deterministic density dependent Ricker population equation to model the dendritic spine dynamics in the neocortex and show that population ecology concepts can provide new insights for short-term spine activity, as well as for the lifetime trajectory of dendritic spines. Some of the testable predictions that emerged from this study are:

In the short-term, high rates of spine generation can cause chaotic oscillations that can lead to over-population or extinction locally.

Small amounts of “immigration” (or a constant stable spine population) can avoid chaos and stabilize the population.

The overall lifetime trajectory of the dendritic spine density can be controlled mainly with three parameters: (i) average spine generation rate, (ii) time-dependent carrying capacity, and (iii) time-dependent filopodia density.

We proposed the use of available metabolic energy supply to model the carrying capacity but one can consider other regulatory mechanisms such as homeostasis in electrical activity and develop the model further.

Finally, we hope that this study will spark interactions between experts in ecology and neuroscience, which can ultimately lead to more accurate models and guide future experimental studies.

AO conceived the theoretical ideas in this work, developed the mathematical model, ran the simulations and took the lead in writing the manuscript. MO reviewed, provided critical feedback and contributed to the writing of the manuscript.

AO was employed by the IBM Corporation and this study was fully funded by IBM Research Almaden. The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AO thanks management support, which enabled this study.