Synaptic Plasticity Rules

The Beta function integral is a special function in mathematics, defined for two positive real numbers $x$ and $y$ as follows:

This integral converges for $x > 0$ and $y > 0$. The Beta function is closely related to the Gamma function, with the relationship given by:

where $\Gamma(n)$ is defined as:

The Beta function often appears in probability and statistics, particularly in the context of the Beta distribution. Its properties make it useful in various applications, including combinatorial problems and the evaluation of integrals.

Dynamic inconsistency refers to a situation in decision-making where a plan or strategy that seems optimal at one point in time becomes suboptimal when the time comes to execute it. This often occurs due to changing preferences or circumstances, leading individuals or organizations to deviate from their original intentions. For example, a person may plan to save a certain amount of money each month for retirement, but when the time comes to make the deposit, they might choose to spend that money on immediate pleasures instead.

This concept is closely related to the idea of time inconsistency, where the value of future benefits is discounted in favor of immediate gratification. In economic models, this can be illustrated using a utility function $U(t)$ that reflects preferences over time. If the utility derived from immediate consumption exceeds that of future consumption, the decision-maker's actions may shift despite their prior commitments. Understanding dynamic inconsistency is crucial for designing better policies and incentives that align short-term actions with long-term goals.

The Shapley Value is a solution concept in cooperative game theory that assigns a unique distribution of a total surplus generated by a coalition of players. It is based on the idea of fairly allocating the gains from cooperation among all participants, taking into account their individual contributions to the overall outcome. The Shapley Value is calculated by considering all possible permutations of players and determining the marginal contribution of each player as they join the coalition. Formally, for a player $i$, the Shapley Value $\phi_i$ can be expressed as:

where $N$ is the set of all players, $S$ is a subset of players not including $i$, and $v(S)$ represents the value generated by the coalition $S$. The Shapley Value ensures that players who contribute more to the success of the coalition receive a larger share of the total payoff, promoting fairness and incentivizing cooperation among participants.

A transfer function is a mathematical representation that describes the relationship between the input and output of a linear time-invariant (LTI) system in the frequency domain. It is commonly denoted as $H(s)$, where $s$ is a complex frequency variable. The transfer function is defined as the ratio of the Laplace transform of the output $Y(s)$ to the Laplace transform of the input $X(s)$:

This function helps in analyzing the system's stability, frequency response, and time response. The poles and zeros of the transfer function provide critical insights into the system's behavior, such as resonance and damping characteristics. By using transfer functions, engineers can design and optimize control systems effectively, ensuring desired performance criteria are met.

Lyapunov Stability is a concept in the field of dynamical systems that assesses the stability of equilibrium points. An equilibrium point is considered stable if, when the system is perturbed slightly, it remains close to this point over time. Formally, a system is Lyapunov stable if for every small positive distance $\epsilon$, there exists another small distance $\delta$ such that if the initial state is within $\delta$ of the equilibrium, the state remains within $\epsilon$ for all subsequent times.

To analyze stability, a Lyapunov function $V(x)$ is commonly used, which is a scalar function that satisfies certain conditions: it is positive definite, and its derivative along the system's trajectories should be negative definite. If such a function can be found, it provides a powerful tool for proving the stability of an equilibrium point without solving the system's equations directly. Thus, Lyapunov Stability serves as a cornerstone in control theory and systems analysis, allowing engineers and scientists to design systems that behave predictably in response to small disturbances.

A time series is a sequence of data points collected or recorded at successive points in time, typically at uniform intervals. This type of data is essential for analyzing trends, seasonal patterns, and cyclic behaviors over time. Time series analysis involves various statistical techniques to model and forecast future values based on historical data. Common applications include economic forecasting, stock market analysis, and resource consumption tracking.

Key characteristics of time series data include:

• Trend: The long-term movement in the data.
• Seasonality: Regular patterns that repeat at specific intervals.
• Cyclic: Fluctuations that occur in a more irregular manner, often influenced by economic or environmental factors.

Mathematically, a time series can be represented as $Y_t = T_t + S_t + C_t + \epsilon_t$, where $Y_t$ is the observed value at time $t$, $T_t$ is the trend component, $S_t$ is the seasonal component, $C_t$ is the cyclic component, and $\epsilon_t$ is the error term.