Motor Learning Mechanism on the Neuron Scale

Based on existing data, we wish to put forward a biological model of motor system on the neuron scale. Then we indicate its implications in statistics and learning. Specifically, neuron’s firing frequency and synaptic strength are probability estimates in essence. And the lateral inhibition also has statistical implications. From the standpoint of learning, dendritic competition through retrograde messengers is the foundation of conditional reflex and “grandmother cell” coding. And they are the kernel mechanisms of motor learning and sensorymotor integration respectively. Finally, we compare motor system with sensory system. In short, we would like to bridge the gap between molecule evidences and computational models. Main Text: Great strides have been made in the research of motor learning (1-5). Until now however, there still exists a gap between existing models and physiological data (6-8). Control models such as the internal forward models (1, 2) and optimal control models (3) are mainly on the module scale. Learning models such as CMAC are difficult for biological implementation (4, 5). Based on existing data and theories, we wish to put forward a quantitative motor learning model on the neuron scale. Inspired by the “self-organization” idea (9), we only make local rules about neuron and synapse, and the neural network will emerge automatically. Moreover, both excitory and inhibitory neurons share the same framework, merely different in details. Motor neurons in this paper include those in the cerebellum, DCN (deep cerebellar nuclei) and basal ganglia, but excluding Pyramid cells in the cerebral motor area. All ci in this paper are constants, and they have different meanings in different paragraphs. Information about sensory memory can be found in our previous work (10). Motor neuron model is as follows (N): N1) 2 | | 1(1 ) c h f c e     , 1 n i i f    , i i i i f a w g  for excitory synapse, and for inhibitory synapse 3 ( ) i i i i f c a w g   3 ( ) i a c  , where f is the soma’s spike frequency or hyperpolarized degree, h is hormone factor, fi is dendritic spike frequency or hyperpolarized degree, ai is sensitivity of membrane channels, wij is the synaptic strength, gi is the presynaptic spike frequency. N1 is inspired by the MP model (11) and Hodgkin-Huxley model (12). N2) 4 i i i da c g a dt   when depolarized , and 5 6 3 ( )( ) i i i da c g c c a dt    when hyperpolarized. Generally speaking, membrane channels become fatigued when a dendrite is depolarized. And they will recover gradually when hyperpolarized (see Fig. 1). In reality, a dendrite has many synapses including both excitory and inhibitory. Therefore gi should be the summarization of all these inputs. The quantity 3 i i b c a   actually means the fatigue of channels. Therefore, postsynaptic potentials of excitory and inhibitory synapses are proportional to the sensitivity and fatigue of membrane channels respectively. And they follow very similar curves (see Fig. 1). In the rest of this paper, synapse is excitory without explicitly indication and motor neuron is called neuron for short. Incidentally, the exponential functions in this model can be easily implemented in physics and biology. Fig. 1. Postsynaptic signal of motor neuron and sensory neuron. Symbol S and N mean cases with and without presynaptic spike respectively. Specially, symbol N in the motor curve can also represent the case when hyperpolarized. And the dotted line means that there is no actual postsynaptic signal. The two curves correspond to motor neuron and sensory neuron respectively, and they are approximately reversal. However, they are caused by different reasons. Specifically, the sensory curve at the bottom is due to the summarization and decay of postsynaptic signal itself. The motor curve however is due to the changes of channels’ sensitivity. Therefore it has a specific upper bound. Moreover, both excitory and inhibitory motor neurons share the same curve at the top. And inhibitory neurons are essential for motor learning. Similar to sensory neuron, S1 actually reflects the spatial summarization of potentials. And the firing frequency f can be viewed as a probabilistic estimate on conditions of features fi (10). However S2 is very different. And physiological evidences support that motor neuron should have a different computational model from sensory neuron. An important difference is that motor neuron can generate dendritic AP (action potential) independent of soma other than EPSP (excitatory postsynaptic potentials) (8). As results, there is no temporal summarization of postsynaptic signal as in sensory neuron. In addition, stimulation of high frequency induces LTD (long term depression) in motor neurons (13), but it induces LTP (long term potentiation) in sensory neurons (14). Since inducing LTP needs the coincidence of soma’s BAP (backpropagating AP) and postsynaptic signals (15), LTD should be due to lacking BAP. This supports S2 in that membrane channels will become fatigue (see Fig. 1). On the other hand, to induce LTP, BAP should be coincidence with hyperpolarization followed by depolarization (16). This supports that membrane channels could recover when hyperpolarized. T-type channels could be the molecular mechanism of N2 (17, 18). These channels can be activated at moderate voltages, but require hyperpolarization to firstly remove inactivation so that the channels can open. Synapse model is as following: S1) LTP: 1 2 ( ) 0 p dw c s h c w dt    , 3 (1 ) 0 p dr c s h r dt    , and 4 log( ) 0 dw c w r dt   when 0 p s  , where h is the hormone factor, r is the synaptic decay rate, 5 p i s c fa  according to N2. S2) LTD:   6 0 1 d d d d dw c s w w dt     ,   7 0 1 d d d d dr c s r r dt     , and Postsynaptic signal