4.2. Neuron model#
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The neuron model used in our Results is the Hodgkin-Huxley-type model of the lobster pyloric rhythm studied by Prinz et al.. The specific implementation we used can be obtained from René [63], along with a complete description of all equations and parameters. We summarize the model below and refer the reader to that reference for more details.
For each cell \(k\), the potential across a patch of area \(A\) and capacitance \(C\) evolves according to the net ionic current through the cell membrane:
where \(I_{\mathrm{ext}}\) is an arbitrary external current, the \(I_i\) describe ion exchanges between a cell and its environment, and \(I_s\) and \(I_e\) describe ion exchanges between different cells. The \(I_{\mathrm{ext}}\) current can be used to represent a current applied by the experimenter, or the inputs from other cells in the network. The voltage \(\tilde{V}\) which an experimenter records would then be some corrupted version of \(V\), subject to noise sources which depend on their experiment.
We generate the external input \(I_{\mathrm{ext}}\) as a Gaussian coloured noise with autocorrelation:
We do this using an implementation[64] of the sparse convolution algorithm [65]. We found that in addition to being more realistic, coloured noise also smears the model response in time and thus reduces degeneracies when comparing models.
Each cell in the model has eight currents through ion channels, indexed by \(i\): one \(\mathrm{Na}^{+}\) current, two \(\mathrm{Ca}^{2+}\) currents, four \(\mathrm{K}^{+}\) currents and one leak current. Each current is modelled as (square brackets indicate functional dependence)
These equations are understood as describing currents through permeable channels with maximum conductivity \(g_{i}^{k}\) (for membrane currents within the same cells) or \(g_{s}^{kl}\) (for synaptic currents between different cells). Conductivities are dynamic: they are governed by the equations for the gating variables \(m\), \(h\) and \(s\) given above. (Some channels do not have inactivating gates; for these, \(h_i\) is set to 1.) The fixed points \(m_\infty\), \(h_\infty\) and \(s_\infty\), as well as the time constants \(τ_m\), \(τ_h\) and \(τ_s\), are functions of the voltage; the precise shape of these functions is specific to each channel type and can be found in either Prinz et al. [6] or René [63].
Following Prinz et al., we treat the functions \(m_\infty\), \(h_\infty\), \(s_\infty\), \(τ_m\), \(τ_h\) and \(τ_s\), as well as the electrical conductance \(g_e\) and the Nernst (\(E_i\)) and synaptic (\(E_{s,k}\)) reversal potentials, as known fixed quantities. Thus the only free parameters in this model are the maximum conductances \(g_{i}^{k}\) and \(g_{s}^{kl}\): the former determine the type of each neuron, while the latter determine the circuit connectivity.
The pyloric circuit model studied by Prinz et al. consists of three populations of neurons[1] with eight different ion channels. Biophysically plausible values for the channel and connectivity parameters were determined through separate exhaustive parameter searches by Prinz et al. [6, 51], the results of which were reduced to sixteen qualitatively different parameter solutions: 5 AB/PD cells, 5 LP cells and 6 PY cells. Importantly, these parameter solutions are distinct: interpolating between them does not yield models which reproduce experimental recordings. For purposes of illustration we study the simple two-cell circuit shown in Fig. 2.1a, where an AB cell drives an LP cell. Moreover we assume the parameters of the AB cell to be known, such that we only need to compare model candidates for the LP cell. (These assumptions are not essential to applying our method, but they avoid us contending with model-specific considerations orthogonal to our exposition.)
The AB neuron is an autonomous pacemaker and serves to drive the circuit with realistic inputs; all of our examples use the same AB model (labelled AB/PD 3 in Table 2 of Prinz et al. [6]), whose output is shown in Fig. 2.1b. Panels c and d show the corresponding response for each of the five LP models given in Table 2 of Prinz et al. [6].
To generate our simulated observations, we use the output of LP 1, add Gaussian noise and then round the result (in millivolts) to the nearest 8-bit integer; in this way \(\Mtrue\) includes both electrical and digitization noise. This leaves LP 2 through LP 5 to serve as candidate models; for these we assume only Gaussian noise, and we label them \(\M_A\) to \(\M_D\). We use \(Θ_a\) to denote the concatenation of all parameters for a given model \(a\), which here consist of the vectors of conductance values \(g_i\) and \(g_s\).
Model definitions are summarized in Table 4.1.
Model |
Model components |
|---|---|
\(\Mtrue\) |
\(I_{\mathrm{ext}} \rightarrow\) |
\(\M_A\) |
\(I_{\mathrm{ext}} \rightarrow\) |
\(\M_B\) |
\(I_{\mathrm{ext}} \rightarrow\) |
\(\M_C\) |
\(I_{\mathrm{ext}} \rightarrow\) |
\(\M_D\) |
\(I_{\mathrm{ext}} \rightarrow\) |
For \(t \in \Tspace\):
Integrate Eq. 4.1 to obtain \(V^{\AB{}}(t)\) and \(V^{\LP{\!\!}}(t; Θ_\true)\).
Draw \(ξ(t) \sim \nN(0, σ_o)\).
Evaluate \(\tilde{V}^{\LP{\!\!}}(t) = \mathtt{digitize}_8\Bigl(V^{\LP{\!\!}}(t) + ξ(t)\Bigr)\).
The \(\mathtt{digitize}_8(x)\) function simulates the data encoding of a digital sensor by converting \(x\) to an 8-bit integer. It does this by first restricting the value of \(x\) to the range \([-128, 127]\), and then rounding towards zero.
For \(t \in \Tspace\) and \(a \in \{A, B, C, D\}\):
Integrate Eq. 4.1 to obtain \(V^{\AB{}}(t)\) and \(V^{\LP{\!\!}}(t; Θ_a)\).
Draw \(ξ(t) \sim \nN(0, σ_o)\).
Evaluate \(\tilde{V}^{\LP{\!\!}}(t) = V^{\LP{\!\!}}(t; Θ_a) + ξ(t)\).
4.3. Loss function for the neuron model#
To evaluate the risk of each candidate model, we use the log likelihood of the observations (Eq. 2.5). This standard choice is convenient for exposition purposes: it is simple to explain and illustrates the generality of the method, since a likelihood function is available for any model in the form of Eq. 1.1.
However it is not a requirement to use the negative log likelihood as the loss, and in fact for time series models it can be disadvantageous. For example, the neuron models used in this work have sharp temporal responses (spikes), which makes the log likelihood sensitive to the timing of these spikes. In practice a less sensitive loss function may be preferable, although the best choice will depend on the application.