Hello, I'm about to join a team working on auditory speech perception using iEEG. It is planned that I will use Temporal Response Function (TRF) to determine correlations between stimulus characteristics (variations in the acoustic signal envelope, for example) and characteristics of recorded neuronal activity.
I would therefore like to fully understand the different stages of data processing carried out, as well as the reasoning and hypotheses behind them.
I took a look at the article presenting the method
Article The Multivariate Temporal Response Function (mTRF) Toolbox: ...
and I studied the matrix calculationsBut several questions remain.
In particular, regarding this formula:
w = (ST S)-1 ST r
where S is a matrix of dimension (T*tau) presenting the characteristics of the stimulus over time (T) as a function of different temporal windows/shifts (tau) as :
S =
[ s(tmin-taumin) ... s(t) ... s(tmin-taumax) ]
[ ... ... ]
[ ... ... ]
[ s(tmax-taumin) ... s(t) ... s(tmax-taumax) ]
and where r is a matrix of dimension (T*N) presenting the recorded activity of each channel in time.
Hypothesis: STS represents the "covariance" of each time window with the others (high covariance in the diagonal (because product of equal columns), high covariance for adjacent columns (because product of close time windows) and low covariance for distant columns whose time windows are very far apart (and therefore presenting little mutual information)). Maybe that (STS)-1ST (of dimension T*tau) makes it possible to obtain a representation of the stimulus according to time windows and time, but with the abrogation of any correlations that may exist between windows? However, the representation of the stimulus in this product remains very unclear to me... And finally, w may represents the weights (or correlations) of each N channel for the different time windows of the signal. My incomprehension mainly concerns the representation of the stimulus by (STS)-1ST and I would like to better understand the reasoning behind these operations and the benefits they bring to the decoding of neural activity. I'd like to thank anyone familiar with TRFs for any help he/she can give me. My reasoning may be wrong or incomplete, any contribution would be appreciated.
Hey Camille, just a bit of help if this works for you.
TRF Analysis Overview:
Formula: w = (STS)^-1STr
1. STS (Covariance Matrix):
- Calculate covariance between different time points in the stimulus.
- Example: Given a stimulus matrix S:
| s1(t=1) s1(t=2) s1(t=3) |
| s2(t=1) s2(t=2) s2(t=3) |
2. (STS)^-1 (Inverse Covariance Matrix):
- Normalize the data by finding the inverse of the covariance matrix.
- Example: If covariance matrix is:
| 10 4 2 |
| 4 8 3 |
| 2 3 7 |
3. (STS)^-1ST (Weighting Matrix):
- Multiply inverse covariance by the transpose of the stimulus.
- Example: Resulting matrix represents the relationship between time windows and neural channels.
4. w (Weight Matrix):
- Represents the weights or correlations between stimulus time windows and neural activity channels.
- Example: If N neural channels, w looks like:
| w1,1 w1,2 ... w1,N |
| w2,1 w2,2 ... w2,N |
TRF analysis helps understand how stimulus variations relate to recorded neural activity, aiding in decoding auditory perception.
Here's a follow up Camille,
Weight Matrix w in TRF Analysis:
The weight matrix w is a fundamental output of Temporal Response Function (TRF) analysis, providing insights into how different aspects of the stimulus relate to neural activity.
Mathematical Representation:
- Each row of w corresponds to a specific time window in the stimulus, denoted as t=1, t=2, t=3, and so on.
- Each column of w corresponds to a neural activity channel, represented as Channel 1, Channel 2, and so forth.
- The values in the weight matrix w are calculated using the formula:
w = (STS)^-1STr
Example:
Suppose we have a simplified weight matrix w, where rows represent different time windows and columns represent neural channels:
| w1, Channel 1 w1, Channel 2 ... w1, Channel N |
| w2, Channel 1 w2, Channel 2 ... w2, Channel N |
| w3, Channel 1 w3, Channel 2 ... w3, Channel N |
In this matrix:
- w1, Channel 1 represents the weight or correlation between the first time window (t=1) of the stimulus and neural Channel 1.
- w2, Channel 2 represents the weight or correlation between the second time window (t=2) of the stimulus and neural Channel 2.
- Each value w captures how strongly a specific time window influences the activity in a particular neural channel.
Interpretation:
- Larger positive values of w indicate that a particular time window has a strong positive influence on the neural activity in a given channel.
- Smaller positive values indicate a positive but weaker influence.
- Negative values suggest a negative correlation, meaning that the time window has an inhibitory effect on neural activity in that channel.
Practical Use:
By examining the weight matrix w, researchers can pinpoint which temporal aspects of the stimulus are most relevant for explaining neural responses. This information is crucial for understanding how auditory stimuli are processed in the brain and aids in the decoding of auditory speech perception.
Transformer-based models, like GPT-3, use Token-level Reasoning Frameworks (TRFs) for various tasks and reasoning processes. TRFs are not a standard term, but I can explain the reasoning behind the use of Transformers in general:
Parallelization: Transformers are highly parallelizable, meaning they can process multiple tokens (words or subwords) in a sequence simultaneously. This parallel processing makes them computationally efficient and allows them to handle long sequences effectively.
Attention Mechanism: Transformers employ a self-attention mechanism, which allows them to weigh the importance of different tokens in the input sequence when generating an output. This attention mechanism enables them to capture complex relationships and dependencies in the data.
Transfer Learning: Pre-trained Transformer models, like GPT-3, are fine-tuned for various downstream tasks. This transfer learning approach leverages the knowledge learned during pre-training and adapts it to specific tasks. This is a key reason behind the success of Transformers in natural language processing.
Scalability: Transformers can be scaled up by increasing their model size and the amount of training data. This scalability has led to the development of larger and more powerful models that achieve state-of-the-art results in various NLP tasks.
Language Agnosticism: Transformers are not limited to specific languages. They can be used for a wide range of languages and are adaptable to multilingual applications, making them versatile for global use.
Contextual Understanding: Transformers excel at capturing contextual information, which is crucial for tasks like language understanding, machine translation, and text generation. They consider the context of a word or token within the entire input sequence, leading to more accurate results.
Adaptability: Transformers can be fine-tuned for different tasks with minimal architectural changes. This flexibility makes them suitable for a wide range of applications, from text classification to language translation to image captioning.
In summary, the reasoning behind the use of Transformers, like GPT-3, is their ability to efficiently handle sequential data, capture complex relationships, support transfer learning, scale to large models, work across languages, and provide contextual understanding for various natural language processing tasks
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Regression