Calcium dependent plasticity applied to repetitive transcranial magnetic stimulation with a neural field model

Abstract

The calcium dependent plasticity (CaDP) approach to the modeling of synaptic weight change is applied using a neural field approach to realistic repetitive transcranial magnetic stimulation (rTMS) protocols. A spatially-symmetric nonlinear neural field model consisting of populations of excitatory and inhibitory neurons is used. The plasticity between excitatory cell populations is then evaluated using a CaDP approach that incorporates metaplasticity. The direction and size of the plasticity (potentiation or depression) depends on both the amplitude of stimulation and duration of the protocol. The breaks in the inhibitory theta-burst stimulation protocol are crucial to ensuring that the stimulation bursts are potentiating in nature. Tuning the parameters of a spike-timing dependent plasticity (STDP) window with a Monte Carlo approach to maximize agreement between STDP predictions and the CaDP results reproduces a realistically-shaped window with two regions of depression in agreement with the existing literature. Developing understanding of how TMS interacts with cells at a network level may be important for future investigation.

Appendices

Appendix

1.1 The neural field model

We use the well-established neural field model of Robinson et al. (1997, 2005). This model uses the average soma potentials, neural firing rates and axonal pulse rates as its key variables. It is nonlinear since the response of a cell to incoming synaptic events is nonlinear. For completeness, we present the model briefly here, before moving on to consider how synaptic plasticity is incorporated (Fung and Robinson 2013; 2014).

We look at the mean soma potential V _a of neurons of type a (a = e or i for excitatory and inhibitory neuronal populations respectively) at a position $\vec {r}$ and time t. The soma potential is a result of contributions from incoming postsynaptic potentials (PSPs):

$$ V_{a}(\vec{r},t)={\sum}_{b} V_{ab}(\vec{r},t), $$

(6)

where $V_{ab}(\vec {r},t)$ is the result of a postsynaptic potential (PSP) of type b onto the cell of type a (a = e or i; b = e, i or x). The dendritic behavior is governed by

$$ D_{ab} V_{ab}(\vec{r},t) = \nu_{ab}(\vec{r},t) \phi_{ab}(\vec{r},t-\tau_{ab}), $$

(7)

where the coupling strength $\nu _{ab}(\vec {r},t)$ is the product of the average synaptic weight $s_{ab}(\vec {r},t)$ and the average number of connections N _{a b} to a cell of type a from a cell of type b, that is: ν _{a b} = s _{a b} N _{a b} , and ϕ _{a b} is the mean axonal pulse rate to a cell of type a from a cell of type b. The term τ _{a b} describes a delay in propagation of the signal to a from b (e.g., introduced by a corticothalamic feedback loop). In general this loop is important for many features of the electroencephalogram especially of order 10–20 Hz (Robinson et al. 1997, 2001). However, when a subject has his or her eyes open, as is the case for most TMS studies, the resonance at around 10 Hz is weak. This allows us to set τ _{a b} =0 for simplicity without losing biophysical relevance. Inclusion of the loop is therefore an important point for future study. In this work we do not consider spatial effects. Therefore we remove reference to the $\vec {r}$ label in what follows. To some extent this is reasonable since a TMS pulse excites a few cm ² of cortex, but we recognize that spatial effects may be important in human TMS (Funke and Benali 2010) and this remains a topic for future study. The operator D _{a b} describes the time evolution of V _{a b} as a result of synaptic input, and is given by

$$ D_{ab}=\left( \frac{1}{\alpha_{ab}} \frac{d}{dt} + 1\right) \left( \frac{1}{\beta_{ab}} \frac{d}{dt} + 1 \right). $$

(8)

where α _{a b} and β _{a b} describe the combined rates of the synaptic and soma dynamics; specifically α _{a b} is the rise rate of the response of a neuron of type a due to synaptic input from neurons of type b and β _{a b} is the decay rate. This is equivalent to the more general propagator form

$$ V_{ab}(t) = \nu_{ab}(t) \int L_{ab}(t-t') \phi_{ab}(t^{\prime})dt^{\prime} , $$

(9)

where L _{a b} is the Green function corresponding to D _{a b} with the assumption that ν _{a b} changes slowly. For this work, we use the differential form of Eq. (8). The average firing-rate of neurons of type a depends on soma potential, with

$$ Q_{a}=Q_{a}^{max}S(V_{a}), $$

(10)

where

$$ S(V_{a})=\frac{1}{1+e^{-(V_{a} - \theta_{a})/\sigma_{a}}}. $$

(11)

To describe propagation of mean signals along populations of axons we model mean axonal pulse rates with the damped wave equation (Robinson et al. 1997)

$$ {\mathcal D}_{ab} \phi_{ab}(t)= Q_{b}(t), $$

(12)

where

$$ {\mathcal D}_{ab} = \left( \frac{1}{\gamma_{ab}} \frac{d}{dt} + 1 \right)^{2} $$

(13)

Note that Eq. (13) in its full form has a Laplacian term describing spatial variation, but this has been removed since we assume there are no spatial variations. This is equivalent to the integral form:

$$ \phi_{ab}(t) = \int \int {\Gamma}_{ab}(t-t^{\prime}) Q_{b}(t^{\prime}) dt^{\prime}, $$

(14)

where Γ _{a b} is the Green function corresponding to ${\mathcal D}_{ab}$, r _{a b} denotes the average range of axons to type a cells from type b cells, and γ _{a b} is an axonal rate constant. We assume that connections between neurons are randomly distributed, so ν _{a b} can be labeled solely by the postsynaptic population, so ν _{a b} = ν _b . Furthermore, we consider the synaptic dynamics as being dominated by the presynaptic cell (that is, the dynamics of the neurotransmitter) and so we can also identify that γ _{a b} = γ _b , α _{a b} = α _b , β _{a b} = β _b , meaning ϕ _{a b} = ϕ _b .

We include multiple inhibitory time-scales for the inhibitory synaptic dynamics (Pérez-Garci et al. 2006) through both GABA _A and GABA _B effects by using two lots of Eq. (7) for the inhibitory cells: $D_{ai}^{\mathrm {A}}$ and $D_{ai}^{\mathrm {B}}$ acting on $V_{ai}^{\mathrm {A}}$ and $V_{ai}^{\mathrm {B}}$ respectively. Their contributions can be added through Eq. (6).

Finally, we add in an external axonal driving stimulation ϕ _x to both the excitatory and inhibitory populations. We alter the strength of stimulation received by the two different populations e and i by modulating ϕ _x by parameters λ _{e x} and λ _{i x} , for the e and i destination populations respectively.

The model is shown in a diagrammatic form in Fig. 9, and the standard parameters for this (taken from references Fung and Robinson 2013 and Fung and Robinson 2014 except for the synaptic coupling terms which have been selected to match the gains used in Wilson et al. 2014) are included in Table 1. The model and parameter set, without a delay term τ _{a b} , cannot produce a strong alpha (≈10 Hz) peak in the spectrum of the firing-rate, but is consistent with the typical ‘eyes open’ approach taken in experimental rTMS in order to remove the complicating effects of an alpha rhythm.

Fig. 9

The populations and connections for the neural field model. Three populations are considered (marked by the gray boxes): an excitatory population e, and inhibitory population i and an external excitatory driving population x. The dotted lines indicate that the synaptic coupling ν _{e e} is dependent on ϕ _e and the excitatory soma potential V _e arising from the addition of the PSPs in Eq. (6)

Table 1 Parameters for the neural field model and calcium dependent metaplasticity model, drawn from references (Fung and Robinson 2013, 2014; Wilson et al. 2014)

Calcium dependent metaplasticity

We now summarize the approach of Fung and Robinson (2014) to model the synaptic changes via CaDP with a metaplasticity scheme. The major effector of plasticity is considered to be the postsynaptic intracellular calcium concentration [Ca²⁺] _a (where a can represent the excitatory or inhibitory population) which is modulated through NMDA receptors (Shouval et al. 2002). Specifically, we can model the ultimate synaptic weight $\tilde {s}_{ab}(t)$ through

$$ \frac{d\tilde{s}_{ab}}{dt} = \eta([\text{Ca}^{2+}]_{a}) \left( s_{max} {\Omega}([\text{Ca}^{2+}]_{a})- \tilde{s}_{ab} \right). $$

(15)

The ultimate synaptic weight $\tilde {s}_{ab}$ describes the value that the synaptic weight will ultimately come to if all external stimulation is stopped — that is, it determines ultimately whether a protocol will show LTP or LTD. However, the actual synaptic weight responds slower than this, as explained below. Equation (15) has a Green function that represents an exponential decay of s _{a b} towards s _{m a x} Ω with a rate parameter η. Both η and Ω depend on postsynaptic concentration of Ca ²⁺; the rate parameter η is low at low Ca ²⁺ concentrations and high at moderate to high concentrations; whereas Ω is approximately 0.5 at low concentrations (<0.15 μM), 0 at medium concentrations (0.15–0.5 μM), and rises to 1 at high concentrations (>0.5 μM), representing little plasticity, LTD, and LTP respectively.

The actual synaptic weight s _{a b} will respond more slowly, for example due to a sequence of protein cascades. We model this through a differential equation

$$ \left( z_{ab} \frac{d}{dt} +1 \right)^{2} s_{ab} = \tilde{s}_{ab}. $$

(16)

where z _{a b} is a characteristic response timescale.

The postsynaptic calcium concentration [Ca²⁺] _a (where a can represent an excitatory or inhibitory cell) itself depends on the glutamate binding and postsynaptic activity through

$$ \frac{ d[\text{Ca}^{2+}]_{a} }{dt} = g_{a} B([\text{glu}]) H(V_{a}) - \frac{[\text{Ca}^{2+}]_{a}}{\tau_{\text{Ca}}}, $$

(17)

where g _a is the NMDA receptor-modulated calcium permeability, B models the extent of glutamate binding through a sigmoid function of glutamate concentration [glu], H represents a voltage-dependent modulation of the dynamics (increasing with V _a except at very high depolarizations), and τ _Ca a time-constant for calcium dynamics.

Metaplasticity is incorporated in a Bienenstock-Cooper-Munro (BCM) scheme (Bienenstock et al. 1982). In the BCM approach, the activity level that demarcates the boundary between LTD and LTP is dependent on previous activity. This can be incorporated into the CaDP plasticity approach by making the calcium conductance g _a dependent upon the history of the weight s _{a b} . Specifically, we can write

$$ \frac{dg_{a}}{dt} = \frac{1}{\tau_{\text{rec}}}(g_{a0} - g_{a}) - \frac{g_{a0}}{\tau_{\text{BCM}}} \left( \frac{\tilde{s}_{ab}}{s_{ab}} -1 \right) $$

(18)

where g _a0 represents the NMDA receptor-modulated calcium conductance at equilibrium, s _{a b} is the actual synaptic weight, and $\tilde {s}_{ab}$ is the ultimate synaptic weight (termed ‘target synaptic weight’ by Fung et al. 2014). There are two timescales here; τ _BCM describes the timescale of the metaplasticity and τ _rec, which is much longer than τ _BCM, is the timescale for calcium conductance to recover to the stable value.

Finally, the glutamate concentration depends on generation by presynaptic activity, with a decay over a timescale τ _glu, as modeled by the equation

$$ \frac{d [\text{glu}]}{dt} = \lambda_{\text{glu}} {\sum}_{E} \phi_{E} - \frac{ [\text{glu}]}{\tau_{\text{glu}}}, $$

(19)

where λ _glu is the glutamate concentration released per presynaptic excitatory spike and ${\sum }_{E} \phi _{E}$ is the incoming synaptic flux summed over all excitatory populations.

Equations (15)–(19), with associated functions η, Ω, B and H (Fung and Robinson 2013), represent CaDP with metaplasticity. The resulting s _{a b} is fed back into Eq. (7) through ν _{a b} = N _{a b} s _{a b} . Equivalently, we can define Eqs. (15), (16) and (18) in terms of the couple ν _{a b} , through $\tilde {\nu }_{ab} = N_{ab} \tilde {s}_{ab}$ and ν _{m a x} = N _{a b} s _{m a x} which is what is done in this work. In this way, we can work with the synaptic couplings $\tilde {\nu }_{ab}$ and ν _{a b} rather than the $\tilde {s}_{ab}$, s _{a b} and N _{a b} .

Parameters and functions for these equations are as defined by Fung and Robinson (2013, 2014); parameters are listed in Table 1.

Deriving an equivalent STDP window

In this section we describe how an equivalent STDP calculation can be carried out assuming a window function H _{e e} (τ). To do this we first linearize the model then transform to Fourier space. This allows us to find the spectra of fluctuations in firing rate δ Q _a (t) and fluctuation in axonal pulse rate δ ϕ _{a b} (t) in response to the external stimulus ϕ _x (t).

Since the neural field model is mostly linear, with Eq. (11) and changes due to plasticity being the only nonlinear contributions, linearizing is straightforward. We can write

$$ Q_{a}(t) = Q_{a}^{\text{eqm}} + \delta Q_{a}(t) , $$

(20)

and

$$ \phi_{ab}(t) = \phi_{ab}^{\text{eqm}} + \delta \phi_{ab}(t) $$

(21)

where $Q_{a}^{\text {eqm}}$ and $\phi _{ab}^{\text {eqm}}$ are the equilibrium cell firing rates and mean axonal pulse rates, respectively. We then use the Fourier form in time. We use the following definition of the Fourier representation of a periodic function x(t) with period T:

$$ x(t) = {\sum}_{n} x_{n} e^{-i \omega_{n} t}, $$

(22)

where ω _n =2π n/T with n an integer, so that

$$ x_{n}=\frac{1}{T} {{\int}_{0}^{T}} x(t) e^{i \omega_{n} t} dt. $$

(23)

This is a useful form to use since we are considering rTMS, where the stimulation is repetitive over a time-scale T. We can then consider x(t) to repeat in a time-scale T, this is reasonable if T is much smaller than the timescale over which weights change. We note that for iTBS, T will be typically 10 s (2 s ON followed by 8 s OFF), whereas the mean weights may vary over a scale of order one minute or longer (Fung et al. 2013; Wilson et al. 2014; Ridding and Rothwell 2007) so such a description is reasonable. The function x(t) and its Fourier representation x _n carry the same dimensions.

We can write the Eqs. (7) and (12) as

$$ \delta Q_{a n} = T {\sum}_{b} G_{ab} L_{ab n} \delta\phi_{abn} $$

(24)

and

$$ \delta \phi_{ab n} = T {\Gamma}_{ab n} \delta Q_{b n} $$

(25)

where

$$ L_{ab n} = \frac{1}{T (1-i\omega_{n}/\alpha_{ab}) (1 - i \omega_{n}/\beta_{ab})}, $$

(26)

$$ {\Gamma}_{ab n} = \frac{1}{T (1 - i\omega_{n}/\gamma_{ab})^{2}}, $$

(27)

and

$$ G_{ab}(t) = \nu_{ab}(t) \left. \frac{\partial Q_{a}}{\partial V_{ab}} \right|_{\text{eqm}} $$

(28)

is the gain, which we have assumed to vary only slowly with time.

In Eq. (24), the sum over the suffix b, denoting the presynaptic population, is taken over the excitatory population e and both GABA _A and GABA _B effects of the inhibitory population i, as shown in the ‘ Σ’ signs of Fig. 9.

Equations (24)–(28) are now simply linear and algebraic. They can be solved with matrix techniques for a given stimulation protocol (Robinson 2011; Wilson et al. 2014).

We now incorporate an STDP description of plasticity into the linearized model. Robinson (2011) and Fung et al. (2013) showed how this can be done, and again for completeness we summarize it here. We define an STDP window function H _{a b} (τ), meaning that a postsynaptic event a time τ after a presynaptic event (for a b-to-a synapse) gives a relative change in synaptic weight of H _{a b} (τ).

We now define a relative synaptic weight w _{a b} such that w _{a b} = ν _{a b} (t)/ν _{a b} (t=0), so that w _{a b} (t=0)=1. The rate of change of relative synaptic weight is then given by (Robinson 2011; Fung et al. 2013)

$$ \frac{d w_{ab}}{dt} = {\int}_{-T/2}^{T/2} \left< \delta Q_{a}(t^{\prime}+\tau) H_{ab}(\tau) \delta \phi_{ab}(t^{\prime}) \right>_{t^{\prime}} d\tau , $$

(29)

where the temporal average denoted by the angle brackets is centered on t and both the integral and the temporal average are taken over a timescale long enough that H _{a b} (τ) is negligible for τ>T/2. The assumption here is that the STDP is pairwise since the integral of Eq. (29) implicitly considers all pairs of presynaptic and postsynaptic events equally, but does not include any consideration of higher-order (e.g., triplet) terms (Shah et al. 2006).

In Fourier space, Eq. (29) can be written as a straightforward sum over all frequencies (Robinson 2011), making its evaluation fast, with

$$ \frac{d w_{ab}}{dt} = T {\sum}_{n} \delta Q_{an}^{*} \delta \phi_{abn} H_{abn}. $$

(30)

The spectra δ Q _{a n} and δ ϕ _{a n} are found for a particular protocol. Then, for a given H _{a b} (τ), we can Fourier transform to find H _{a b n} and use the sum in Eq. (30) to quickly find the rate of change of weight as predicted by pairwise STDP. Although it is quick to evaluate Eq. (30), if the change in synaptic weight is fed back into the calculation through the excitatory-to-excitatory gain, calculations must be performed over discrete time steps so some of the efficiency is lost. However, since the weights change much more slowly than the firing rates and mean axonal pulse rates, we can use large time steps for updating the synaptic weight. In this work, we concentrate on the initial rate of change of synaptic weight and therefore do not feed the weight back into the calculation.