Optogenetics are a process of neurology circuit investigations:

1. every neuron which expresses as specific change becomes sensitive to light
2. therefore, you can shine a light on the mouse’s brain to control it

This uses a set of molecules named opsins.

Recall: Oracle Turing Machines is a machine which has oracle \(B \subseteq \Gamma^{*}\) which lets you ask “if \(z \in B\), then do something, otherwise do something else” and check that in ONE STEP.

We have:

\begin{equation} P^{B} = \qty {L \mid L \text{{ can be decided by a polynomial-time TM with oracle for $B$ }}} \end{equation}

For example \(P^{\text{SAT}}\) are languages we can decide in polynomial time once we have an oracle in SAT, and \(P^{\text{NP}}\) are languages decidable by some Oracle Polynomial Time \(TM\) for some \(B \in NP\).

Oracle Turing Machine

An Oracle Turing Machine \(M\) is a machine that can ask membership queries in a set \(B \subseteq \Gamma^{*}\) on a special “oracle tape”. This Turing machine will receive an answer in one step: in particular, if the string written on the oracle tape is in \(B\), then the state will be set to \(q_{yes}\), and otherwise, we move to \(q_{no}\)

Importantly, \(B\) does not have to be decidable. We think of them as subroutines.

oral lexical retrival is a class of discourse tasks which asks the subject to convert some semantic understanding (“concept”) into lexical expressions (“words”)

“ask a patient to describe a thing.”

Examples of oral lexical retrieval:

• SVF
• BNT

Source: CambridgeCore

ODEs are Differential Equations in one independent variable: \(y(x)\).

Main Content:

• First-Order Differential Equations
• Second-Order Linear Differential Equations
• Uniqueness and Existance

Overarching Categories

order of equations

the order of an equation is the highest derivative of an equation

linear vs. non-linear differential equations

A solution of a differential equation is linear when solutions are closed under linear operations.

We can spot an ODE by seeing that each of its derivatives are seperated or in separable terms, and only to the first power—because that ends up being a linear equation (i.e. any two solutions satisfying the equation can add and scale to another solution).