Simplifying Dendritic trees into a single quadratic current.

What is the science 

Neurons have large intricate and complex branching dendritic trees, that act as the input into the neuron. These large complex trees are responsible for filtering and integrating the input. Because dendritic trees receive inputs in specific branches of the dendritic tree they can perform spatial filtering on the inputs. This allows the neuron to be selective to direction, to perform coincidence detection, and perform integration on the synapses.

However these effects have been traditionally thought of to require the dendritic tree. If one were to remove the dendritic tree, then the neuron could not perform these functions. When building computer models of neural networks, often modelers leave the dendritic tree off of the  model. Thus the neurons will lose these abilities.


 What did they do

However Dr. Songting Li, Dr. Nan Liu, Dr. Xiaohui Zhang, Dr. David W. McLaughlin, Dr. Douglas Zhou, and Dr. David Cai, recently tried to see if they could use a new equation to recover the loss of functionality from not modelling the dendritic tree. They used two-port theory form electronics to derive the transfer function for excitatory and inhibitory synapses. One way to think about transfer functions, is it represents the amount of signal attenuation that a particular synapse will receive on the way from its source on the dendritic tree, to the soma. It also can incorporate some of the resulting nonlinear interactions from the other synapses.

What did they find

One can see that this is indeed the case for by examining equation 5 in the paper.
$$ I_{syn}=\sum_m \sum_n \alpha^{m n}_{E E} g^m_E  g^n_E +\sum_p \sum_q \alpha^{p q}_{E I} g^p_E  g^q_I+\sum_s \sum_t \alpha^{s t}_{I I} g^s_I g^t_I $$
note here, that the interaction of multiplying terms $g_E$ and $g_I$ gives rise to the nonlinear interaction seen in the transfer function. Here the term $\alpha$ is a scalar value derived by the transfer function. Note the sum runs over every possible combintation of two synapses, including the synapse interacting with itself.  This new synaptic current is now quadratic, because it mixes $g_E*g_I$ together. With this new synaptic current defined, the authors are able to reproduce several functions that dendrites are capable of performing, including direction selectivity, coincidence detection and, synaptic integration.

 What's the impact

Ideally this methodology will allow simulations of large neural networks to add functionality into their individual neurons. These new quadratic synaptic currents will allow time consuming dendritic simulations to be replaced by a shorter quadratic equation that is easier to simulate large neural networks. This will require less memory and also speed up the simulations compared to the default multi-compartmental models currently used.


Author: Alexander J. White
Source Dendritic computations captured by an effective point
neuron model https://www.pnas.org/content/116/30/15244