## Math451H Applied Mathematics课程简介

Applied mathematics is the mathematical study of general scientific concepts, principles, and phenomena that, because of their widespread occurrence and application, relate or unify various disciplines. The core of the program at MIT concerns the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.

Sophomores interested in applied mathematics typically survey the field by enrolling in 18.200 and 18.300 Principles of Applied Mathematics. Subject 18.200 is devoted to the discrete aspects of the study and may be taken concurrently with 18.03. It carries CI-M credit in mathematics. Subject 18.300, given only in the second term, is devoted to continuous aspects and makes considerable use of differential equations.

The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.

Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.

## PREREQUISITES

• 18.03 or 18.032 (formerly 18.034) (Differential Equations)
[sufficiently advanced students may substitute 18.152 or 18.303]
• 18.04 (Complex Variables with Applications)
or 18.112 (Functions of a Complex Variable)
• 18.06, 18.C06 (formerly 18.061), 18.700 or 18.701 (Linear Algebra)
• 18.200 (Principles of Discrete Applied Mathematics)
or the non-CI-M variant 18.200A
• 18.300 (Principles of Continuum Applied Mathematics)

## Math451H Applied Mathematics HELP（EXAM HELP， ONLINE TUTOR）

Consider the following passive membrane equation
$$C \frac{d V}{d t}=-G_L\left(V-E_L\right)+I_{a p p}+I_{i n}(t)$$
where $V$ is the membrane potential $(\mathrm{mV}), t$ is time measured in msec , $C$ is the membrane capacitance $\left(\mu \mathrm{F} / \mathrm{cm}^2\right), I_{a p p}$ is the applied bias (DC) current $\left(\mu \mathrm{A} / \mathrm{cm}^2\right), I_{\text {in }}(t)$ is a timedependent input current $\left(\mu \mathrm{A} / \mathrm{cm}^2\right)$,

Write a code to solve eq. (1) using the modified Euler method (Runge-Kutta, order 2) with time step $\Delta t=0.1$.
Consider the following parameter values: $E_L=-65, G_L=0.1, C=1$

1. Plot the numerical solution to eq. (1) for $I_{i n}(t)=0$ and
(a) $I_{a p p}=0.5$.
(b) $I_{a p p}=1$.

To solve the passive membrane equation using the modified Euler method, we first need to discretize the equation in time. We can use the following update rule:

V_{n+1} = V_n + \frac{\Delta t}{2C} \left[-G_L\left(V_n-E_L\right) + I_{a p p} + I_{\text{in}}(t_n) – G_L\left(V_{n}-E_L\right) + I_{a p p} + I_{\text{in}}(t_{n+1})\right]Vn+1​=Vn​+2CΔt​[−GL​(Vn​−EL​)+Iapp​+Iin​(tn​)−GL​(Vn​−EL​)+Iapp​+Iin​(tn+1​)]

where $V_n$ is the membrane potential at time step $n$, $t_n=n\Delta t$, and $t_{n+1}=(n+1)\Delta t$. Note that we use the midpoint approximation for the input current $I_{\text{in}}$.

Now, we can write the Python code to solve the equation and plot the results:The code uses a sinewave input current with a period of 10 ms. We can see that the membrane potential oscillates in response to the input current. The amplitude and frequency of the oscillations depend on the strength and frequency of the input current. The plots for $I_{a p p}=0.5$ and $I_{a p p}=1$ are shown below.

1. Plot the numerical solution to eq. (1) for $I_{a p p}=1$ and
$$I_{\text {in }}(t)=I_{a p p} \operatorname{Heav}\left(t-t_i\right) * \operatorname{Heav}\left(t_f-t\right)$$
where $\operatorname{Heav}(t)$ is the Heaviside function, with
(a) $t_i=100$ and $t_f=200$.
(b) $t_i=100$ and $t_f=400$.

To plot the numerical solution to eq. (1) for a step input current, we need to modify the input current function to implement the Heaviside function. We can do this using NumPy’s heaviside function, which returns the Heaviside function value for an array of inputs.

The updated Python code to solve the equation and plot the results is:

The code defines the step input current using the Heaviside function and solves the equation using the modified Euler method. The plot for $I_{a p p}=1$ and $t_i=100$ and $t_f=200$ is shown below.

The membrane potential increases quickly in response to the step input and then decays back to the resting potential over time. The amplitude and duration of the response depend on the strength and duration of the input current. The plot for $t_i=100$ and $t_f=400$ looks similar, but the response lasts longer.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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