Application and Computation of Probabilistic Neural Plasticity

Soaad Hossain

arXiv:1907.00689·q-bio.NC·August 10, 2020

Application and Computation of Probabilistic Neural Plasticity

Soaad Hossain

PDF

Open Access

TL;DR

This paper introduces a mathematical framework using differential equations and spike-train statistics to compute and infer probabilistic neural plasticity, aiming to enhance understanding across neuroscience and AI.

Contribution

It presents a novel approach to model and compute neural plasticity probabilistically using additive short-term memory equations and differential equations.

Findings

01

Formulation of an additive STM equation for neural plasticity

02

Application of probabilistic inference to neural plasticity modeling

03

Potential interdisciplinary applications in AI, psychiatry, and behavioral science

Abstract

The discovery of neural plasticity has proved that throughout the life of a human being, the brain reorganizes itself through forming new neural connections. The formation of new neural connections are achieved through the brain's effort to adapt to new environments or to changes in the existing environment. Despite the realization of neural plasticity, there is a lack of understanding the probability of neural plasticity occurring given some event. Using ordinary differential equations, neural firing equations and spike-train statistics, we show how an additive short-term memory (STM) equation can be formulated to approach the computation of neural plasticity. We then show how the additive STM equation can be used for probabilistic inference in computable neural plasticity, and the computation of probabilistic neural plasticity. We will also provide a brief introduction to the theory…

Equations62

Δ x_{i} = (Δ x_{i})_{in t er na l} + (Δ x_{i})_{e x c i t a t or y} + (Δ x_{i})_{inhibi t or y} + (Δ x_{i})_{s t im u l i}

Δ x_{i} = (Δ x_{i})_{in t er na l} + (Δ x_{i})_{e x c i t a t or y} + (Δ x_{i})_{inhibi t or y} + (Δ x_{i})_{s t im u l i}

(Δ x_{i})_{in t er na l} = (\frac{d x _{i} ( t )}{d t})_{in t er na l} = - A_{i} (x_{i}) x_{i}

(Δ x_{i})_{in t er na l} = (\frac{d x _{i} ( t )}{d t})_{in t er na l} = - A_{i} (x_{i}) x_{i}

\mathrm{\Delta}w_{i}=\left\{\begin{array}[]{ll}A_{+}exp\left\{\frac{\mathrm{\Delta}t_{i}}{\tau_{+}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }<0\\ {-A}_{-}exp\left\{\frac{-\mathrm{\Delta}t_{i}}{\tau_{-}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }>0\end{array}\right.

\mathrm{\Delta}w_{i}=\left\{\begin{array}[]{ll}A_{+}exp\left\{\frac{\mathrm{\Delta}t_{i}}{\tau_{+}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }<0\\ {-A}_{-}exp\left\{\frac{-\mathrm{\Delta}t_{i}}{\tau_{-}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }>0\end{array}\right.

\left(\mathrm{\Delta}x_{i}\right)_{inhibitory}\ =\left(\frac{dx_{i}(t)}{dt}\right)_{inhibitory}=\mathrm{\Delta}w_{i}\

\left(\mathrm{\Delta}x_{i}\right)_{inhibitory}\ =\left(\frac{dx_{i}(t)}{dt}\right)_{inhibitory}=\mathrm{\Delta}w_{i}\

\mathrm{\Delta}w_{i}=\left\{\begin{array}[]{ll}A_{+}exp\left\{\frac{\mathrm{\Delta}t_{i}}{\tau_{+}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }<0\\ {-A}_{-}exp\left\{\frac{-\mathrm{\Delta}t_{i}}{\tau_{-}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }>0\end{array}\right.

\mathrm{\Delta}w_{i}=\left\{\begin{array}[]{ll}A_{+}exp\left\{\frac{\mathrm{\Delta}t_{i}}{\tau_{+}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }<0\\ {-A}_{-}exp\left\{\frac{-\mathrm{\Delta}t_{i}}{\tau_{-}}\right\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \mathrm{\Delta}t_{i\ }>0\end{array}\right.

\int_{0}^{T} \frac{[ \frac{d s ( t )}{d t} ]}{T} = 0

\int_{0}^{T} \frac{[ \frac{d s ( t )}{d t} ]}{T} = 0

s (T + τ) = s (τ)

s (T + τ) = s (τ)

C (τ) = [\frac{1}{n} \sum r (t \textsubscript i - τ)] \approx [\frac{1}{< n >} \sum s (t \textsubscript i - τ)]

C (τ) = [\frac{1}{n} \sum r (t \textsubscript i - τ)] \approx [\frac{1}{< n >} \sum s (t \textsubscript i - τ)]

C (τ) = \frac{1}{< n >} \int_{0}^{T} \frac{d g [ \sum _{i = 1}^{n} t - t _{i} ]}{d t} s (t + τ)

C (τ) = \frac{1}{< n >} \int_{0}^{T} \frac{d g [ \sum _{i = 1}^{n} t - t _{i} ]}{d t} s (t + τ)

\int_{0}^{T} \frac{d h [ s ( t + τ ) ]}{d t} = \int_{τ}^{T + τ} \frac{d h [ s ( t ) ]}{d t} = \int_{0}^{T} \frac{d h [ s ( t ) ]}{d t}

\int_{0}^{T} \frac{d h [ s ( t + τ ) ]}{d t} = \int_{τ}^{T + τ} \frac{d h [ s ( t ) ]}{d t} = \int_{0}^{T} \frac{d h [ s ( t ) ]}{d t}

(Δ x_{i})_{s t im u l i} = (\frac{d x _{i} ( t )}{d t})_{s t im u l i} = \int_{0}^{T} \frac{d h [ s ( t + τ ) ]}{d t}

(Δ x_{i})_{s t im u l i} = (\frac{d x _{i} ( t )}{d t})_{s t im u l i} = \int_{0}^{T} \frac{d h [ s ( t + τ ) ]}{d t}

P (t_{1}, t_{2}, ..., t_{n}) = p (t_{1}, t_{2}, ..., t_{n}) (Δ t)

P (t_{1}, t_{2}, ..., t_{n}) = p (t_{1}, t_{2}, ..., t_{n}) (Δ t)

P (t_{1}, t_{2}, ..., t_{n}) = n! P_{T} (n) (\frac{Δ t}{T})^{n}

P (t_{1}, t_{2}, ..., t_{n}) = n! P_{T} (n) (\frac{Δ t}{T})^{n}

p (t_{1}, t_{2}, ..., t_{n}) = i = 1 \prod n r (t_{i}) e x p {\int_{0}^{T} \frac{d r ( t )}{d t}}

p (t_{1}, t_{2}, ..., t_{n}) = i = 1 \prod n r (t_{i}) e x p {\int_{0}^{T} \frac{d r ( t )}{d t}}

(Δ x_{i})_{e x c i t a t or y} = (\frac{d x _{i} ( t )}{d t})_{e x c i t a t or y} = ρ_{i}

(Δ x_{i})_{e x c i t a t or y} = (\frac{d x _{i} ( t )}{d t})_{e x c i t a t or y} = ρ_{i}

\rho_{i}=\left\{\begin{array}[]{ll}n!P_{T}\left(n\right)\left(\frac{\Delta t}{T}\right)^{n}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ the\ firing\ rate\ is\ constant\ over\ time\\ \prod_{i=1}^{n}r\left(t_{i}\right)exp\left\{\int_{0}^{T}\frac{dr\left(t\right)}{dt}\right\}\ \ \ \ if\ the\ firing\ rate\ is\ time-dependent\end{array}\right.

\rho_{i}=\left\{\begin{array}[]{ll}n!P_{T}\left(n\right)\left(\frac{\Delta t}{T}\right)^{n}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ the\ firing\ rate\ is\ constant\ over\ time\\ \prod_{i=1}^{n}r\left(t_{i}\right)exp\left\{\int_{0}^{T}\frac{dr\left(t\right)}{dt}\right\}\ \ \ \ if\ the\ firing\ rate\ is\ time-dependent\end{array}\right.

Δ x_{i}

Δ x_{i}

= - A_{i} (x_{i}) x_{i} + ρ_{i} + Δ w_{i} + \int_{0}^{T} \frac{d h [ s ( t + τ ) ]}{d t}

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

P (Y ∣ T, D N P)

P (Y ∣ T, D N P)

= P [Y ∣ D N P (T_{1} = t_{1}), D N P (T_{2} = t_{2}), ..., D N P (T_{n} = t_{n})]

P (D N P ∣ E, T) = \frac{P ( E ∣ D N P , T ) P ( T ∣ D N P ) P ( D N P )}{P ( D N P ∣ E , T ) P ( E ∣ T ) P ( T )}

P (D N P ∣ E, T) = \frac{P ( E ∣ D N P , T ) P ( T ∣ D N P ) P ( D N P )}{P ( D N P ∣ E , T ) P ( E ∣ T ) P ( T )}

P (D N P ∣ E, T) = \frac{P ( E ∣ D N P , T ) P ( T ∣ D N P )}{P ( D N P ∣ E , T ) P ( E ∣ T )}

P (D N P ∣ E, T) = \frac{P ( E ∣ D N P , T ) P ( T ∣ D N P )}{P ( D N P ∣ E , T ) P ( E ∣ T )}

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

P (D N P ∣ E, T)

P (D N P ∣ E, T)

= \frac{P ( A c t i o n _{p a r e n t s} ∣ D N P , T ) P ( T ∣ D N P ) P ( D N P )}{P ( D N P ∣ A c t i o n _{p a r e n t s} , T ) P ( A c t i o n _{p a r e n t s} ∣ T ) P ( T )}

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

Δ x_{i}

Δ x_{i}

= - A_{i} (x_{i}) x_{i} + i = 1 \prod n r (t_{i}) e x p {\int_{0}^{T} \frac{d r ( t )}{d t}} + - A_{-} e x p {\frac{- Δ t _{i}}{τ _{-}}} + \int_{0}^{T} \frac{d h [ s ( t + τ ) ]}{d t}

P (D N P ∣ E, T)

P (D N P ∣ E, T)

= \frac{P ( O f f e n s e _{v i o l e n ce} ∣ D N P , T ) P ( T ∣ D N P ) P ( D N P )}{P ( D N P ∣ O f f e n s e _{v i o l e n ce} , T ) P ( O f f e n s e _{v i o l e n ce} ∣ T ) P ( T )}

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

D N P = M o d e ([Δ x_{i}]) = M o d e ([Δ x \textsubscript 1, Δ x \textsubscript 2, ..., Δ x \textsubscript n])

Δ x_{i}

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Taxonomy

TopicsModel Reduction and Neural Networks · Neural dynamics and brain function · Neural Networks and Applications

Full text

Application and Computation of Probabilistic Neural Plasticity

Soaad Q. Hossain

Department of Computer and Mathematical Sciences, Department of Philosophy

University of Toronto, Scarborough

Toronto, Canada

[email protected]

Abstract

The discovery of neural plasticity has proved that throughout the life of a human being, the brain reorganizes itself through forming new neural connections. The formation of new neural connections are achieved through the brain’s effort to adapt to new environments or to changes in the existing environment. Despite the realization of neural plasticity, there is a lack of understanding the probability of neural plasticity occurring given some event. Using ordinary differential equations, neural firing equations and spike-train statistics, we show how an additive short-term memory (STM) equation can be formulated to approach the computation of neural plasticity. We then show how the additive STM equation can be used for probabilistic inference in computable neural plasticity, and the computation of probabilistic neural plasticity. We will also provide a brief introduction to the theory of probabilistic neural plasticity and conclude with showing how it can be applied to behavioural science, machine learning, artificial intelligence and psychiatry.

1 Introduction

Changes in the functionality and development of the brain has and continues to be an ongoing discussion and research within the neuroscience, psychology, artificial intelligence, cognitive science, and psychiatric community. Discoveries about the brain have led to major realizations about it, shifting how we act, learn, perceive and approach situations and circumstances. The realization of neural plasticity is one of those realizations about the brain that created a drastic shift in our understanding of the brain as it showed us that the development of the brain does not stop at adulthood; rather it continues to change throughout a person’s entire life. In other words, changes to the brain can take place from infancy to adulthood.

While the realization took place many years later, the use of it took place in the 20th century by neuroscientist and pathologist Santiago Ramón y Cajal. He described neuronal plasticity as the nonpathological changes in the structure of the brain of an individual (Fluchs & Flügge, 2014). Arguments and evidence overtime proved the existence of neural plasticity, leading to the current debates and discussions that we have about it today. Understanding the probability of neural plasticity occuring given some event is a topic of interest, but has not been adequately covered or discussed. To contribute to this discussion, we will propose and briefly introduce the theory of probabilistic neural plasticity, then derive a short-term memory (STM) equation that can be used for the computation of probabilitistic neural plasticity, elaborate how the STM equation can be used for probabilistic inference in computable neural plasticity. In addition, we will also review relevant work on neural plasticity. We will conclude a discussion on the application of probabilistic neural plasticity in areas such as behavioral science, machine learning, and psychiatry.

2 Preliminaries

In understanding changes within the brain, we define several terms within neuroanatomy. Neurons are the cells within the nervous system that communicate with each other to transmit information to other nerve cells, muscles, or gland cells. Neurons generally contain an axon, a cell body, and dendrites. The cell body contains the nucleus and cytoplasm. The axon is part of the cell body, extending from the cell body to the nerve terminals. They often create smaller branches prior to ending at the nerve terminals. Dendrites extend from the cell body and receive messages from other neurons within the brain. They are composed synapses formed by the ends of the axon from other neurons.

2.1 Synapses

Synapses are the spaces found within the brain cells (Bate & Bassiri, 2016). Historically synapses were generally understood to carry the role of transferring information between one neuron to another through connecting them together. However, it was later discovered that the efficacy of synaptic transmission is not constant; that it varied depending on the frequency of the stimulation, and the modulation of synaptic frequency provoked modifications of neural connections in volume, size and shape (Bate & Bassiri, 2016). In other words, those connections that are frequently stimulated will increase in volume and size, resulting in a change in shape. Consequently, inactive connections decrease in size and volume, resulting in synaptic long-term depression. The creation of synapses is known as synaptic formation and elimination of synapses is known as synaptic pruning. While synaptic formation and pruning occur extensively during the early stages of brain development (i.e. embryonic brain stage), synaptic formation and pruning takes place throughout until the end stages of brain development (i.e. adult brain). The adult brain continues to create synapses and eliminate unwanted synapses over the course of our life in response to our actions do and experiences, playing important roles in learning, memory, and other aspects of the functioning of our brain. Such evolution of nervous systems enables us to adapt to environments and determine the optimal action in any given situation based on what was learned from our past experiences (Constandi, 2016). As a result, synapses have their efficacy reinforced or weakened as a function of experience. The occurrence of modulation of synaptic frequency provokes modifications of neural connections in volume, size and shape is known as neural plasticity, synaptic plasticity, or neuroplasticity.

2.2 Stimuli

Stimuli are defined as events or occurences in the environment of an organism that influences its behavior. In general, neurons responding to sensory stimuli face the challenge of encoding parameters that can vary over a vast dynamic range (Dayan & Abbott, 2001). To address such wide-ranging stimuli, sensory neurons often respond most strongly and rapid changes in stimulus properties and are relatively insensitive to steady-state levels. Mathematically, the functions of stimulus intensity are typically with logarithmic or weak power-law dependence (Dayan & Abbott, 2001).

2.3 Plasticity

The term plasticity can be defined as a structure that is weak enough to submit to influence, but strong enough to not submit to all influence at once (Constandi, 2016). From this definition, if we consider the brain as a structure, the term neural plasticity can be understood as influences causing a modification of neural connections with changes occurring over a specific span of time. The type of plasticity that we are concerned with is functional plasticity. Functional plasticity involves changes in some physiological aspect of nerve cell functions such as frequency of nervous impulse or the probability of release of chemical signals - both of which act to make synaptic connections either stronger or weaker, resulting in changes to the degree of synchronicity among populations of cells. All types of plasticity that the brain can undergo enables the brain to learn from experience in order to form memories and acquire new skills, and also allows for adaptation and recovery from brain trauma, or at least to compensate for and work around any damage that has occurred. Consequently, the relationship between brain and behavior is not one-sided – experiences and behaviors induce plastic changes in the brain, which these changes can influence our future behavior and experiences (Constandi, 2016).

3 Related work

Work intersecting probability theory and neural plasticity have investigated neural plasticity through using a Bayesian approach and probabilistic inference, resulting in the development of models. However, there are no formal theories that fully combine the two. The only theory that would be considered most closely related to these would be the theory of probabilistic neural network (PNN) (Specht, 1989). This theory is derived from Bayesian computing and Kernal density method (Sebastian et al., 2019), and only focuses on artificial neural networks (ANN) (Mohebali et al., 2020). Aside from PNN, there is one work that is closely related that investigates neurons and behavior with respect to Bayesian inference. Darlington et al. have described a complete neural implementation of Bayesian-like behavior that includes an adaptation of a prior. Their work focused on addressing a specific aspect of neurons - whether it is possible to decode the behavioral effects of context and stimulus contrast from the neural population (Darlington et al., 2018).

There are some studies within the area of machine learning that propose models involving neural plasticity and probability theory. In a study conducted by Tully et al., they investigated the a Hebbian learning rule for spiking neurons inspired by Bayesian statistics. They found that that neurons can represent information in the form of probability distributions and that probabilistic inference could be a functional by-product of coupled synaptic and nonsynaptic mechanisms operating over multiple timescales (Tully et al., 2014). Similarly, in a study by Kappel et al., they proposed that inherent stochasticity enables synaptic plasticity to carry out probabilistic inference, and presented a model for synaptic plasticity that allows spiking neural networks to compensate continuously for unforeseen disturbances (Kappal et al., 2015). These models help with establishing the connection between neurons and probability distributions, and synapses and probabilistic inference.

4 Probabilistic neural plasticity

Within plasticity, the way that probability is involved comes from the influence. More specifically, when we consider a structure that is weak enough to submit to influence, but strong enough to not submit to all influence at once, there are two probabilities that can be discussed. The first probability that can be discussed is the probability that the influence can cause a structure to be influenced. The second probability that can be discussed is the probability that a structure is strong enough to not submit to all influence at once. This tells us that probability plays a major role in neural plasticity. Through understanding the role that it plays in neural plasticity; we can understand the chance of neural plasticity occurring given some event. To comprehend the probability that of neural plasticity occurring given some event, we define and refer to the theory of probabilistic neural plasticity.

Definition: Probabilistic neural plasticity is the probability that specific effects are due to causes that are defined by variations of neural connections formed by the brain from adapting to its environment.

In this context, effects are either a behavioral condition or a mind condition. The environment consists of not only the physical environment, but also events, situations and circumstances. For instance, a traumatic event can be considered as an environment which can cause the brain to undergo neurological changes needed for it to adapt to the traumatic event. Alternatively, an environment can also be a data set as a data set used for training and testing the machine learning algorithm can influence the behavior and performance of the algorithm. What can be noticed about probabilistic neural plasticity is that its application can be found in both human learning and in machine learning. We will further discuss about its application after discussing the computational components associated with probabilistic neural plasticity and the computation of probabilistic neural plasticity.

To perform the computation of probabilistic neural plasticity, we must formulate computational and mathematical models using theories and concepts from areas such as theoretical neuroscience, mathematics, and computer science. Furthermore, probabilistic inferences can be made through the computational aspect of neural plasticity. Moving forward, we will discuss and elaborate on the computation of neural mechanisms, covering neural state equations, neural firing, spike-train statistics, and the neural plasticity equation. We will then discuss about probabilistic inference in computable neural plasticity followed by the computation of probabilistic neural plasticity.

5 Computable neural mechanisms and equations

In approaching the computation of neural mechanisms and the computation of neural plasticity, we will describe a computational model of neural systems. More specifically, we will describe the computational model of neuron states, neural firing, and spike-train statistics. Between physiological, psychological and neurological approaches, while the computational (and mathematical) modeling can be done using any one of those approaches, we will be approaching neural mechanisms and neural plasticity solely through a neurological approach. In using theoretical neuroscience and additive short-term memory (STM) equation, we will formulate the neural plasticity equation and a computational model for neural plasticity.

5.1 Neural state equations

The computational and mathematical model of neuron states can be written as neuron state equations. Assume a neuron interacting with other neurons and external stimuli. Let vi be defined as the ith neuron. Let xi and wi be defined as the variable that describes the neuron’s state. Let $\tau$ be defined as time interval. Then the one state variable of vi is xi where xi is denoted as the activation level of the ith neuron. The second state variable, wi, is associated with vi’s interaction with vj’s (another neuron). Using additive STM equation we formulate the set of ordinary differential equations for xi and wi. Assume a change caused by internal and external processes in neuron potential from equilibrium, and that inputs from other neurons and stimuli are additive (agrees with many experiments). Then for all i = 0, 1, 2, … , n, the additive STM equation is

[TABLE]

The additive STM equation is the $\Delta$ xi neuron state equation that describes the activation level of xi with respect to internal, excitatory, inhibitory and stimuli. We will describe neural firing through the relationship between stimulus and firing-rate, and we will address the internal ordinary differential equation, excitatory ordinary differential equation, inhibitory ordinary differential equation, and stimuli ordinary differential equation. We will then formulate the neural plasticity equation needed for the computation of probabilistic neural plasticity.

To obtain the internal ordinary differential equation in the additive STM equation, we assume that the neuron processes are stable. In doing so, this provides us the internal ordinary differential equation

[TABLE]

with Ai(xi) $>$ 0. A is denoted as the learning rates for the potentiation.

For the inhibitatory ordinary differential equation in the additive STM equation, it can be obtained through the change in the neuron’s state. With wi being defined as the variable that describes the neuron’s state, the change in the neuron’s state can be described using $\Delta$ wi. To obtain the equation for $\Delta$ wi, we can make use of existing mathematical models. A mathematical model that we can make use of is the mathematical model Bi-Phasic Spike Timing Dependent Plasticity (Bi-Phasic STDP). Bi-Phasic STDP consists of two phases. The first phase is a depressive phase in which pre- synaptic spike follows post-synaptic spike. The second phase is a potentiating phase in which a post-synaptic spike follows pe-synaptic spike (Chrol-Cannon, and Jin, 2014). The Bi-Phasic STDP model is

[TABLE]

In the model, A+ is denoted as the learning rates for the potentiation and A- is denoted as the learning rates for the depression, $\Delta$ ti is denoted as the delay of the post-synaptic spike occurring after the transmission of the pre-synaptic spike, and $\tau$ + and $\tau$ - controls the rates of the exponential decrease in plasticity across the learning window. Using the Bi-Phasic STDP model, we obtain the inhibitory ordinary differential equation

[TABLE]

which

[TABLE]

5.2 Stimulus

To recall, when neurons encounter wide-ranging stimuli, sensory neurons often respond most strongly and rapid changes in stimulus properties and are relatively insensitive to steady-state levels. Steady-state responses are highly compressed functions of stimulus intensity. Let s be denoted as a stimulus. Applying Weber’s law tells us that when we differentiate between the intensity of two stimuli, $\Delta$ s is proportional to the magnitude of s, resulting in $\Delta$ s/s being constant. In terms of adaptation, sensory systems make many adaptations, using a variety of mechanisms to adjust to the average level of stimulus intensity (Dayan, & Abbott, 2001). When a stimulus generates such adaptation, describing responses to fluctuations about a mean stimulus level is an ideal way of comprehending the relationship between the stimulus and the response (Dayan & Abbott, 2001). In this case, s(t) be defined such that its time average over the duration of a trial is 0. Then, we find that

[TABLE]

We can now use two approaches that we can use to progress in analyzing stimulus. The first approach would be to use several different stimuli and averaging over them. The second approach is putting all the stimuli that we wish to consider into a single time dependent stimulus sequence and average over time. We will use the second approach, replacing stimulus averages with time averages. To make the stimulus periodic, for any time $\tau$ , we define the stimulus outside the time limits of the trial by the relation

[TABLE]

5.3 Neural firing

For neural firing, we define spike-triggered average stimulus as C( $\tau$ ), where C( $\tau$ ) is denoted as the average value of the stimulus a time interval $\tau$ before a spike is fired. Mathematically, we define the spike-triggered average stimulus as

[TABLE]

The approximate equality of the mathematical equation comes from the fact that if n is large, then the total number of spikes on each trial is well approximated by the average number of spikes per trial (Dayan, and Abbott, 2001). Expressing the spike-triggered average stimulus as an integral would be

[TABLE]

In formulating the stimuli ordinary differential equation in the additive STM equation, we use the approach that consists of placing all of the stimuli that we want to consider into a single time-depeendent stimulus sequence and average over them - replacing stimulus averages with time averages. Then, for any h, the integrals involving the stimulus being time-translationally invariant is

[TABLE]

Then the stimuli ordinary differential equation is

[TABLE]

5.4 Spike-train statistics

In approaching the relationship between a stimulus and a response on a stochastic level, we use probability theory to address and approach spike tunes and occurences, and neuronal firing. For spike tunes, as they are continuous variables, the probability for a spike to occur at a specific time is 0. In turn, to obtain a nonzero spike value, we must evaluate the probability that a spike occurs within a specific time interval (Dayan & Abbott, 2001). Let the probability that a spike occurs in the time interval between times t and t + $\Delta$ t. Given such time interbal, the probability P(t1,t2, …, tn) that a sequence of n spikes occurs with spike i falling between time interval ti and ti + $\Delta$ t for i = 1, 2, …, n is given in terms of density by the relation of

[TABLE]

To obtain an approximation of stochastic neuronal firing, Poisson process can be used as Poisson process entails that events are statistically independent if they are not dependent at all on preceding events (Dayan & Abbott, 2001). In using Poisson process, we can approach firing process through two separate cases. The first case is a homogeneous process that involves the firing rate being constant over time (Dayan, & Abbott, 2001). In this case, since the firing rate is constant, the Poisson process generates every sequence of n spikes over a fixed time interval with equal probability. Let PT(n) be the probability that an arbitrary sequence of exactly n spikes occurring within a trial of duration T. Let the firing rate for a homogeneous Poisson process be denoted asq(t) = q) (as it is independent of time). Assume that the spike times are ordered. Then 0 $\leq$ t1 $\leq$ t1 $\leq$ … $\leq$ tn $\leq$ T and the probability for n spike times is

[TABLE]

The second case is a inhomogeneous poisson process that involves the firing rate being time-dependent (Dayan, & Abbott, 2001). In this case, since the firing rate depends on time, different sequences of n spikes occur with different probabilities. In addition, the probability p(t1,t2 …, tn) depends on the spike times. Using inhomogeneous Poisson process, we find that the spikes are still generared independently, and their times enter p(t1,t2 …, tn only through the time-dependent firing rate r(t). Assume that the spike times are ordered. Then 0 $\leq$ t1 $\leq$ t1 $\leq$ … $\leq$ tn $\leq$ T and the probability for n spike times is

[TABLE]

For the excitatory ordinary differential equation in the additive STM equation, we assume additive synaptic excitation proportions to the spike-train frequency. We find that for the excitatory ordinary differential equation, the equation depends on the firing rate and whether it is constant over time or time-dependent. This gives us the ordinary differential equation

[TABLE]

where

[TABLE]

5.5 Neural plasticity equation

Combining all of the ordinary differential equations together, for all i = 0, 1, …. n, the neuron state equation is

[TABLE]

which $\rho$ i and wi are defined with respect to their ordinary differential equations. To compute neural plasticity, we consider all of the activities of the neurons from 1 to n. The activity level that most frequently occurred among all the neurons is the neuron state with has the highest degree of neural plasticity. The neuron state that has the highest degree of neural plasticity therefore has the most influence on the brain, directly contributing to the (plastic) changes occurring inside the brain. The final computational model that can be used for neural plasticity, which we will name the Degree of Neural Plasticity Model (DNP model) would then become the following: let xi and wi be defined as the variables that describe the neuron’s state. Then for all i = 0, 1, …. n

[TABLE]

In the model, we denote $\Delta$ wi as the change in synaptic strength; wi as the synaptic strength; A+ as the learning rate for the potentiation; A- as the learning rate for the depression; $\tau$ + and $\tau$ - as what controls the rates of exponential decrease across the learning window; T as trial; and xi as the activation level of the ith neuron. This establishes the computation of neural plasticity.

6 Probabilistic inference in computable neural plasticity

With establishing the computation of neural plasticity, we can discuss how probabilistic inferences can be made through the computation of neural plasticity. Probabilistic inference is used to approach probabilistic queries of form P $(X|Y)$ , where Y and Z, are disjoint subsets of X given a graphic model for X. We will go into the construction of the graphical model. Rather, we will directly define a probabilistic query and address the probabilistic queries using probabilistic inference. However, note that a basic graphical model can be constructed using the concept of neuron state equations and the Bi-Phase STDP model. In understanding neural plasticity, our query would be to know what the posterior distribution of degrees of neural plasticity would look like. Let Y be the query variable. Let T be denoted as a time interval [ti, t] with i = 1, 2, …, n. Let DNP be denoted as the degrees of neural plasticity. The probabilistic query can address using is

[TABLE]

This would enable us to obtain the posterior probabilities of degree of neural plasticity. This in turn, can then be used to estimate the posterior distribution with respect to degrees of neural plasticity.

7 Computation of probabilistic neural plasticity

To recall, the problem that is being answered is what is the probability that event X will cause neural plastic changes to the brain. The answer to that question can be obtained through using probability neural plasticity - the probability that specific effects are due to causes that are defined by variations of neural connections formed by the brain from adapting to its environment. With the DNP model, we can now investigate and analyze how exactly the computation of probabilistic neural plasticity can be achieved. The first thing we will do is formulate the problem in mathematical terms. Let E be denoted as a specific event Let T be denoted as a trial with [ti, t] with i = 1, 2, …, n. Let DNP be denoted as the degrees of neural plasticity. The problem in mathematical terms would be P $(DNP|E,T)$ . The variable T was added as the time interval must be defined in order to compute the change in synaptic strength. The probabilistic neural plasticity model that would be

[TABLE]

Since T must always be known, P(T) = 1. For P(DNP), since it relies on trial T, we cannot compute for P(DNP) directly. To address this, we can either assume that P(DNP) = 1 or P(DNP) = 0, which for this case we will always assume that P(DNP) = 1. The only time when P(DNP) = 0 is when $\Delta$ ti = 0. This simplifies probabilistic neural plasticity model P $(DNP|E,T)$ to

[TABLE]

Set DNP as the function with respect to trial T. Then

[TABLE]

which this can be solved using the STM equation. From here, we can compute the probability of whether stimuli S will most likely cause neural plastic changes to the brain. If a high probability is obtained from the computation of P $(DNP|S,T)$ , then that implies that stimuli S will most likely cause neural plastic changes to the brain. If a low probability is obtained from the computation of P $(DNP|S,T)$ , then that implies that stimuli S will not most likely cause neural plastic changes to the brain.

8 Application of probabilistic neural plasticity

Our theoretical results stems from behavioral science and machine learning. We wish to study the behavior of humans and artificial intelligence (AI), understanding what is the probability of neural plasticity occuring given some event. For humans, the event would be a stimulus, while for artificial intelligence, the event would be data. In both cases, we wish to understand how likely it is for a human or machine learning algorithm to change or develop a new behavior from experiencing some event. Another way of seeing it is that we want to understand how likely it is for a human or machine learning algorithm to change its thought and reasoning process after experiencing some event. The cases presented are not actual cases that have taken place. Rather, they are used to illustrate how probabilisitic neural plasticity can be applied to behavioral science, machine learning, and psychiatry. In addition, it is important to note that while our dicussion focuses on results that involve behavioral science, machine learning and psychiatry, discussion on the results (and even the application of probabilistic neural plasticity) can futher be elaborated, extending and involving other disciplines such as neuroscience and cognitive science.

8.1 Behavioral science and psychiatry

Within the context of behavioral science, we consider a case in psychiatry. A patient diagnosed with borderline personality disorder is presented. It is provided that the patient expressed chronic feelings of emptiness, and has a fear of abandonment. In addition, the patient display feelings of anger, but does not provide adequate details about their relationship with their parents, making it difficult to determine whether the cause of the chronic feelings of emptiness and fear of abandonment is due to the patient being neglected, withheld, uncared, or abandoned by the caregivers (American Psychiatric Association, 2013). To determine whether neglect from the parents caused neural plastic changes to the brain of the patient, we apply probabilistic neural plasticity. Let DNP be defined as the degree of neural plasticity with respect to the behavior and feelings presented by the patient (chronic feelings of chronic feelings of emptiness, fear of abandonment, and anger). Let E be defined as the parents action towards the patient, and let T be the time interval. Then the equation (20) becomes

[TABLE]

We set the DNP as a function with respect to trial T. Then

[TABLE]

Let n be the current age of the patient. We expand equation (16) with $\rho$ i being defined with the firing rate being time-dependent, and $\Delta$ ti $\geq$ 0. Then the equation for i = 1, 2, …, n, $\Delta$ xi is

[TABLE]

If the probability P $(DNP|E,T)$ $\geq$ 0.5, then neglect from the parents caused neural plastic changes to the brain of the patient. If P $(DNP|E,T)$ $<$ 0.5, then neglect from the parents did not cause neural plastic changes to the brain of the patient.

8.2 Machine learning and artificial intelligence

Within the context of machine learning, we consider an ANN. An ANN designed to predict the probability that an individual that has committed a crime in the past will commit another crime in the future. We know that the data used to train the ANN was provided by the department of justice of the country. The data included sex (male, female), age at release (18 and older) and race/hispanic origin (white, black/African American, hispanic/latino, others), commitment offense (violent, property, drug, and public order), and recidivism. We also know that based on the data, previously convicted males of age range 18 - 29 of race others that commit an offense of violence are most likely to recommit the act of violence. The ANN is being used on a white male, aged 24, that has previously been convicted of violence to determine the probability that that invidivual will commit the same crime again in the next 3 years. The AI (that uses the ANN) predicts that the individual will commit the same crime in the next 3 years. To determine whether the commitment offense data caused neural plastic changes to the ANN to determine whether the individual will recommit the crime in the next 3 years, we apply probabilistic neural plasticity. Let DNP be defined as the degree of neural plasticity with respect to the individual’s criminal record. Let E be defined as the criminal offense, and let T be the time interval. Then the equation (20) becomes

[TABLE]

We set the DNP as a function with respect to trial T. Then

[TABLE]

Let n be the number of years. We expand equation (16) with $\rho$ i being defined with the firing rate being constant over time, and $\Delta$ ti $<$ 0. Then the equation for i = 1, 2, …, n, $\Delta$ xi is

[TABLE]

If the probability P $(DNP|E,T)$ $\geq$ 0.5, then the commitment offense in the data caused neural plastic changes to the ANN, resulting in the decision made by the AI on the previously convicted individual. If P $(DNP|E,T)$ $<$ 0.5, then then the commitment offense data did not cause neural plastic changes to the ANN, resulting in the decision made by the AI on the previously convicted individual.

9 Discussion

In attempt to study the behavior of humans and AI, understanding what is the probability of neural plasticity occuring given some event. For humans, the event would be a stimulus, while for artificial intelligence, the event would be data. In both cases, we wish to understand how likely it is for a human or machine learning algorithm to change or develop a new behavior from experiencing some event. Accordingly, to address this, we proposed probabilistic neural plasticity, performed the computation of it, then applied it to behavioral science, psychiatry, machine learning and AI. In proposing and developing the theory of probabilistic neural plasticity, this contributes to the field of neuroscience and artificial intelligence, and brings us a step closer to answering questions and provide explanations pertaining to learning and behavior, especially within humans. Future work on this theory can focus on collecting empirical evidence to better establish it, ellaborating on the theory itself, and applying the theory in areas such as cognitive science, psychology and deep learning.

10 Acknowledgement

We would like to thank Linbo Wang (University of Toronto) and Mark Fortney (University of Toronto) for useful discussions and suggestions. This manuscript has been released as a pre-print at arXiv (Hossain, 2019).

References

American Psychiatric Association (2013). Diagnostic and Statistical Manual of Mental Disorders. American Psychiatric Publishing. 5, 664.

Bates, D., & Bassiri, N. (2016). Plasticity and Pathology. Berkeley Forum in the Humanities. 27 – 28.

Chrol-Cannon, J., & Jin, Y. (2014). Computational Modeling of Neural Plasticity for Self-Organization of Neural Networks. Biosystems 125, 43 – 54.

Constandi, M. (2016). Neuroplasticity. The MIT Press Essential Knowledge Series. 2, 8 – 13, 28 – 29, 45 – 47.

Dayan, P., & Abbott, L. F. (2001). Theoretical Neuroscience – Computational and Mathematical Modeling of Neural Systems. The MIT Press. 17 – 39, 47 – 51.

Darlington, T. R., Beck, J. M., & Lisberger, S. G. (2018). Neural Implementation of Bayesian Inference in a Sensorimotor Behavior. Nature Neuroscience. DOI: 10.1038/s41593-018-0233-y

Fuchs, E., & Flügge, G. (2014). Adult Neuroplasticity: More Than 40 Years of Research. Hindawi Publishing Corporation. 1.

Hossain, S. Application and Computation of Probabilistic Neural Plasticity. arXiv:1907:00689.

Kappal, D., Habenschuss, S., Legenstein R., & Maass, W. (2015). Synaptic Sampling: A Bayesian Approach to Neural Network Plasticity and Rewiring. In Proceedings of the 28th International Conference on Neural Information Processing Systems (NeurIPS 2015). 370 - 378.

Mohebali, B., Tahmassebi, A., Meyer-Baese, A., & Gandomi, A. H. (2020). Probabilistic Neural Networks. Handbook of Probabilistic Models. 347–350. DOI: 10.1016/b978-0-12-816514-0.00014-x

Sebastian, A., Pannone, A., Subbulakshmi Radhakrishnan, S., & Das, S. (2019). Gaussian Synapses for Probabilistic Neural Networks. Nature Communications. 10(1). DOI: 10.1038/s41467-019-12035-6

Specht, D. F. (1990). Probabilistic neural networks. Neural Networks. 3(1), 109–110. DOI: 10.1016/0893-6080(90)90049-q

Tully, P. J., Hennig, M. H., & Lansner, A. (2014). Synaptic and Nonsynaptic Plasticity Approximating Probabilistic Inference. Frontiers in Synaptic Neuroscience. 6(8), 1 - 16. DOI: 10.3389/fnsyn.2014.00008