Question: Does 'g' in equation 5 denote the softplus?
Evidence: Neurons in the C. elegans nervous system are largely non-spiking, so we model the voltage dynamics for these neurons as a passive point neurons with a single electrical compartment. Let v ∈ R N denote the voltages of the N neurons. The voltage v i (t) for each post-synaptic neuron i at the time t was calculated using a first-order leaky integrator equation given by
where τ i is the voltage time constant, o i is the chemosensory input provided to only the sensory neurons, s c i is the chemical synaptic input, s e i is the electrical synaptic input, v rest i is the resting neuron voltage.
We studied two variations of the model, the current-based model and the conductance-based model, which differ in their formulations of chemical synaptic input s c i . Since neurons in the C. elegans nervous system are largely non-spiking, we model the chemical synapses as having graded release of neurotransmitter, rather than the all-or-none quantal release seen in spiking neurons. In both models, we model the amount of neurotransmitter released, W c ji g(v j (t)), in proportion to the pre-synaptic voltage v j followed by a softplus activation g(•) which sets a minimum voltage below which there is no synaptic release. We use g(•), to denote a softplus function for the rest of the paper. In our current-based model, synaptic input s c i to a post-synaptic neuron i is directly proportional pre-synaptic neurotransmitter concentration:
where W c ji represents the chemical synaptic weight between pre-synaptic neuron j and post-synaptic neuron i. W c ji can be positive or negative depending on if the synaptic connection is excitatory or inhibitory. If neurons j and i are not connected, W c ji is set to zero. In the conductance-based model, we model the synaptic current entering the post-synaptic neuron with more biophysical detail as
Here, the pre-synaptic neurotransmitter concentration W c ji g(v j (t)) is more accurately modeled as proportional to the conductance at the post-synaptic terminal. The post-synaptic current is then given by the product of the synaptic conductance and the difference between the post-synaptic voltage v i (t) and the synaptic reversal potential E ji .
In contrast to the current-based synapse, whose input is independent of the post-synaptic voltage v i (t), the more biophysically accurate conductance-based synapse model also has a dependence on the post-synaptic voltage. Additionally, this model decouples the sign of the synapse from the strength. The reversal potential of a synapse E ji dictates whether a synapse is excitatory or inhibitory. A large and positive E ji corresponds to an excitatory synapse causing depolarization of the postsynaptic neuron. Conversely, an inhibitory synapse will have a negative E ji causing hyperpolarization. In this model, we can now independently train the sign of a synapse and its non-negative strength W c ji , which is not easily possible with current-based synapses. In both the current-based and conductance-based models, the following equation was used to represent electrical synaptic inputs:
where W e ji is restricted to be non-negative and v j − v i is the potential difference between presynaptic and postsynaptic neurons. We also restrict W e ji = W e ij because the potential differences between electrical synapses are symmetric. To directly compare the outputs of the LVM to neural activity measurements, our model must generate calcium signals from the voltage traces. We model the calcium concentration [Ca] i of each neuron i as a first-order leaky integrator, driven by voltage-gated calcium channels with the same nonlinear current-voltage (I-V) function g(v i ):
where g(•) represents softplus activation, τ [Ca] is a time constant shared across all neurons. We then map calcium concentration [Ca] into the measured calcium fluorescence signals f via an affine transform with scalar α f and bias β f ,
with measurement noise represented by a noise amplitude σ f and a noise term f i (t) ∼ N (0, 1).These equations are simulated in discrete time using Euler integration.
where g(·) represents softplus activation,
Answer: ‘g’ represents a softplus with a fixed threshold of 20 in equation 5. The function of equation 5 is the same across neurons. All ‘g’ in our paper represents a softplus activation with a fixed threshold 20. We have included clarifications in our revision.