# Recent papers – LTD and R-F neurons

I am going to keep tabs on the interesting papers I read. I’m actually stealing this idea from Al who uses Google Documents to take notes on papers so they are always at hand!

One of the papers is Izhikevich’s 2001 paper on resonate-and-fire neurons. The most common neural simplification is to use an integrate-and-fire model. This is fine, except some dynamics may be lost. Most importantly, since integrate-and-fire models are 1D, they can only lose stability via a codimension-1 bifurcation (ie, saddle-node). This ensures that certain phenomena, such as spiking in response to inhibitory input, cannot be a part of the model.

The other paper is by Jo and Cho et al. (2008) which we read for the systems and synaptic journal club. I didn’t actually go to the journal club, but I did read the paper! Basically, the authors relate mechanisms for two seperate forms of LTD. One acts through NMDA receptors and the other through mGluR’s. These forms of LTD and seperate, so one can be induced on top of the other to increase the depression.

Click through the link if you care more about these papers (ie, if you are me).

Izhikevich’s resonate-and-fire simplification is a generalized reduction of a two-dimensional system with hopf bifurcation. He relates it to the Young (1937) model which is

$C\dot{V} = I - g_{leak}(V - E_{leak}) - W$

$\dot{W} = (V - V_{1/2})/k - W$

Izhikevich suggests (in his green book) that when V reaches some threshold, it is reset to its reset value a la an integrate-and-fire neuron. This equation is the same as

$\dot{z} = (b + i\omega)z + I$

where z is an imaginary value, b < 0 and $\omega$ > 0. The imaginary component of z is considered ‘voltage-like’ and has a spiking threshold at 1. The figure below shows why it may be useful to use a r-a-f neuron instead of an i-a-f neuron (this type of behavior can be seen experimentally, depending on the cell). One thing it illustrates is that the r-a-f neuron does not have a fixed threshold; i-a-f can be modified to have a variable threshold as well, but the original model does not.

Our other paper concerns synapses in slices of rat perirhinal cortex. The authors identify two distinct forms of LTD, which can occur at the same synapse and utilize different signaling mechanisms. Both rely on activation of postsynaptic glutamate receptors which lead to a rise in postsynaptic Ca2+ levels.

The two forms of LTD are induced by altering the frequency of afferent stimulation; 1Hz stimulation induces NMDAR-LTD and 5Hz stimulation induces mGluR-LTD. By using low doses of the calcium chelator BAPTA (0.2 mM), they find that mGluR-LTD is completely blocked while NMDAR-LTD is not. Therefore, the two forms of LTD have differing calcium dependence. Also, the two are distinct and therefore can be activated sequentially, as seen below:

The bulk of the paper deals with the boring (ie, important) problem of figuring out the signalling pathways. NMDAR-LTD requires calmodulin while mGluR-LTD requires PKC. mGluR further requires NCS-1 which interacts with PICK1 (which interacts in some way with AMPAR; I don’t understand the mechanisms here).

References:

Jo et al. (2008). Neuron. “Metabotropic glutamate receptor-mediated LTD involves two interacting Ca2+ sensors, NCS-1 and PICK1.”

Izhikevich (2001). Neural Networks. “Resonate-and-fire neurons.”