I have read that in a deflationary spiral, lower prices cause banks to increase their interest rates. Is this true and what is the logic behind it?
I suspect there is a small mistake in your notes.
Deflation does not cause banks to increase their interest rates. However it is true that a deflationary spiral (or plain deflation for that matter) causes real interest rates to increase. These are the interest rates that matter for the economy anyway, which is why in macroeconomics we often refer to the real interest rate even if it's not explicitly stated so.
The nominal interest rate, set by banks in part, is the interest rate in terms of money. It determines how much money we have to pay back for credit. The real interest rate is the rate in terms of goods and services. It determines how many goods and services we have to pay back for credit.
The real interest rate is defined as:
$ (1+r) = (1 + i) / (1 + \pi) $, which can be approximated by: $r = i - \pi$, where r is the real interest rate, i is the nominal interest rate set by banks and $\pi$ is the inflation rate.
When we have deflation we get a negative $\pi$, which, as can be seen from the equations above, increases the real interest rate r. The smaller $\pi$ is, the bigger r gets. When $\pi$ is negative, even if we set $i$ to its lowest possible value, which is 0, the real interest paid on it will be positive. This means the amount paid back in goods and services is larger than the amount borrowed and increasingly so the bigger the deflation (smaller $\pi$) is.
To understand the situation you have to realize that there are two types of money in the economy.
Government base money & bank money (mostly deposits).
Bank deposits great outnumber government base money (MB).
When deflation happens, is conceptually very similar to a bank run. Depositors demand short term assets or base money instead of long term assets. Because banks operate by mismatching long term assets with short term liabilities, this result in an inevitable liquidity crunch. Put another way during deflation, bank money is being destroyed.
The banking system is very interconnected. Most banks constantly rely on short term lending from other banks to sustain their maturity mismatching. Imaging in an economy there is 20,000 in bank deposits and 10,000 in base money. If 5,000 is withdrawn that leaves 15,000 deposits to cover 5000 in base money. The ratio changed from 2 to 1 and is now 3 to 1. Banks are even more desperate for reserves to hold off insolvency so will pay more for base money than before. This is why interest rates rise during a deflationary spiral.
Now....the Fed attempts to control the fed funds rate so in theory you don't see this happen (well you do see it in that you see massive acquisitions from the Fed in an effort to keep the banking system solvent).
Conventional interest rate equations do not apply to banks because banks are unique. They create their own liquidity and even their own money.