4. Information Theory

Information is the reduction of uncertainty

1) for equiprobable states uncertainty grows with the number of states

2) Uncertainty should be continuous in prob of states

3) Branching property

A measure of uncertainty about a system's state, describing its capacity to carry information. The formula is:

where p(Ri)p(Ri​) is the probability of a particular outcome in a set of possibilities.

Measures the amount of information shared between two variables, such as a stimulus (S) and a response (R), quantifying how much knowing one reduces uncertainty about the other.

1. Spike Timing: Refers to the precise timing of individual spikes in a neuron, which can encode information.

   • This allows for encoding temporal patterns and sub-millisecond precision in neural responses.

   Rate Coding: Involves the firing rate (number of spikes over time) to represent information.

   • It captures overall neural activity, but may not fully utilize the precision of individual spike timings.

   Both mechanisms can complement each other, with rate coding being used for long-term trends and timing coding used for finer, high-precision details.

• Efficient Coding Hypothesis: Suggests that the brain maximizes the information transmission rate while minimizing energy consumption and other costs.

• Neural systems are thought to optimize their responses (e.g., firing patterns) to carry as much relevant information as possible about the external world.

• In terms of coding efficiency, the brain filters out redundant or unimportant signals and focuses on the most relevant stimuli.

• Efficient Coding: Large monopolar cells are optimized to maximize information transfer while minimizing energy use.

• Energy Efficiency: They encode visual information in an energy-efficient manner, focusing on motion and contrastrather than redundant data.

• Natural Stimuli: These cells are adapted to process natural visual stimuli (such as motion), which are most relevant for the fly’s survival.

• Filtering Redundant Data: They help filter out unnecessary details to maintain focus on essential visual information (e.g., detecting movement).

• Adaptation to Stimulus Changes: Neural systems adjust their responses to accommodate changes in the stimulus distribution, ensuring efficient encoding of relevant information.

• Receptive Field Adjustment: Neurons re-tune their receptive fields to focus on the most important aspects of the new stimuli.

• Efficient Coding: The system maximizes information transfer and minimizes redundancy, even as stimulus statistics change.

• Information Maximization: The neural system adapts by adjusting firing rates or spike timings to maintain high coding efficiency.

• Mathematical Model: Fairhall et al. used information theory to model neural adaptation, emphasizing the minimization of uncertainty and maximizing information.

• Natural vs. Artificial Stimuli: Neural systems process natural stimuli (e.g., environmental sounds) more efficiently than artificial stimuli (random or synthetic signals).

• Efficient Coding: The brain is tuned to the statistical regularities in natural sounds, making neural encoding of these stimuli more efficient.

• Neural Adaptation: Sensory systems adapt to the properties of the environmental stimuli, optimizing information transmission and reducing redundancy.

• Key Insight: Neural systems are optimized for natural stimuli, aligning with the efficient coding hypothesis to maximize information transfer while minimizing energy use.

Meaning:

• EE = Coding efficiency (how effectively the system encodes information).

• I(R;S)I(R;S) = Mutual information (amount of information the response carries about the stimulus).

• H(R)H(R) = Entropy of the response (uncertainty in the response).

Interpretation: Coding efficiency reflects how much of the variability in the neural response is used to convey relevant information about the stimulus. High efficiency indicates optimal encoding with minimal redundancy.

• Receptive Fields: Neurons in the visual cortex respond to specific areas of the visual field.

• Simple Cells: Respond to specific orientations and edges (e.g., vertical or horizontal lines).

• Complex Cells: Respond to motion and more complex stimuli.

• Columnar Organization: Neurons are organized in columns that respond to similar features (e.g., orientation, spatial frequency).

• Feature Detectors: Neurons act as detectors for basic visual features (e.g., lines, edges, motion).

• Hierarchy of Processing: The visual system processes visual information hierarchically, starting with basic features and building into more complex representations.

• Efficiency in Feature Detection: Receptive fields are structured to detect essential visual features, such as edges, lines, and motion, critical for survival and navigating the environment.

• Hierarchical Processing: Simple cells detect basic features (like edges), while more complex cells integrate them to process higher-level patterns (e.g., objects or faces).

• Maximization of Information: The form of receptive fields maximizes the system's ability to extract and transmitrelevant information efficiently, while minimizing redundancy.

• Natural Stimulus Statistics: Receptive fields are tuned to detect patterns that are common and important in the natural environment (e.g., edges, contours, motion).

• Evolutionary Adaptation: The form of receptive fields has evolved to optimize sensory processing for survival, improving the ability to detect and react to the most important environmental cues.

• Distributed Representation: Information is encoded by the collective activity of many neurons, not just a single neuron.

• Tuning Curves: Neurons are tuned to specific features of the stimulus (e.g., orientation, motion direction).

• Firing Rates: Firing rates across a population of neurons encode information about stimulus intensity and other features.

• Sparse Coding: In some cases, only a small subset of neurons is active, improving efficiency.

• Decoding: The brain decodes the population activity patterns to extract meaningful information about the stimulus.

• Applications: Population coding is used in sensory systems (e.g., vision, hearing) and motor control to represent complex information efficiently.

• Sparse Coding: A method of representing an image using a small number of active features (dictionary elements) rather than encoding every pixel.

• Image Patches: The image is divided into small patches, and each is represented by a few basis functions (e.g., edges, textures).

• Infotaxis: C. elegans uses the infotaxis model to maximize information acquisition while foraging for food. It adjusts its behavior based on odor detection and non-detection, updating its likelihood of finding food.

• Exploration vs. Exploitation: The worm switches between exploration (searching new areas) and exploitation (staying in food-associated areas) based on the information gathered.

• Random Walk: Starts with a random walk to explore, then adapts to move more directly toward food as more information is acquired.

• Sensory Input: Uses chemosensory neurons to detect food-related odors and navigate toward food sources.

• Behavioral Studies: Researchers track the movement patterns of C. elegans in controlled environments to understand its foraging strategies and decision-making processes.

• provides a rigorous quantitative description of experimental observations

• formulates rigorous predcitions about not yet observed phenomena

• provides a unifying perspective on diverse systems