**Time**

I

The secret Creaking of hips while journeying Faces of sand wrapped in thick cloaks Dates from the Hereafter sealed in the far end of fruit A glimpse of something

A blink of an eye Then resurrection Things they see with their eyes shut Things they may recognise with their senses and Edges The severity of sleep As they hallucinate Then an awakening It is the time of the tree of the unexpected Befalling them Stomping on arid routes like a raging beast Ravaging the thing guarding all things In a pale of doubts and amulets It is far Farther than the stitch of sound to itself

Is it not, then, the creation of farness?


Excitatory synapses onto interneurons are known to express either LTP or LTD via activation of calcium-permeable glutamate receptors (Kullmann and Lamsa, 2007). Cell-type specific rules have been identified in a study examining five common interneuron subtypes, as defined by axonal projections and molecular expression profiles (Nissen et al., 2010). In this work, the authors found that excitatory synapses express LTP onto PV+ basket cells and LTD onto bistratified cells. Both of these phenomena were independent of N-Methyl-D-aspartate (NMDA) receptor activation and potentially act to shift the inhibition on excitatory cells from the dendrites to the soma. A closer examination of the FF and feedback (FB) excitatory inputs onto PV+ basket cells found that NMDA receptors are only found at synapses with FB afferents, leading to a narrower frequency tuning of LTP at these inputs than at FF inputs (Le Roux et al., 2013). Given the importance of FF inhibition to ensure the temporal fidelity of pyramidal cell firing (Pouille and Scanziani, 2001) and the very tight timelock of basket cell interneurons and pyramidal cells during sharp wave ripples (Klausberger and Somogyi, 2008), it is possible the different properties of LTP at FF and FB synapses is permitting PV+ cells to modulate their activity in accordance with excitation.

PV+ basket cells in area CA1 have recently been shown to undergo a long-term increase in excitability in response to brief high frequency stimulation of SC inputs (Campanac et al., 2013). It was elegantly shown that this enhanced FF inhibition in area CA1 was due to an increase in the inherent excitability of PV+ cells resulting from activation of mGluR5 and subsequent downregulation of D-type potassium current carried by Kv1 channels, termed LTP-IE*PV*−*BC*. The authors demonstrated that clustered spiking in the γ-range was increased, allowing for the speculation that this plasticity may provide a use-dependent modulation of hippocampal γ-oscillations, or even potentially allow for a modulation of the phase lag of PV+ basket cells during θ-activity.

PV+ interneurons may be playing an interesting role in mediating the ability of CA3 neurons to excite CA2 pyramidal cells. A very strong FF inhibition at the SC-CA2 synapse induces a very large hyperpolarization in CA2 pyramidal neurons and completely prevents SC axons from driving firing in CA2 pyramidal neurons (Chevaleyre and Siegelbaum, 2010; Kohara et al., 2014). Upon closer examination, this inhibition was found to undergo an iLTD in response to 10, θ-burst, and 100 Hz stimulus protocols. Furthermore, this iLTD was mediated entirely by the activation of delta opioid receptors, resulting in a lasting decrease in GABA release. Furthermore, by using optogenetics to elicit an IPSC from PV+ interneuron terminals, it was demonstrated that PV+ interneurons are responsible for this plasticity (Piskorowski and Chevaleyre, 2013). Given that CA2 pyramidal neurons express multiple factors that inhibit post-synaptic LTP at the SC-CA2 synapse (Zhao et al., 2007; Simons et al., 2009; Lee et al., 2010), this pre-synaptic iLTD in PV+ cells may be the major mechanism by which the excitability of the SC-CA2 synapse is modulated.

Changes in PV+ cell plasticity during development is thought to underlie the "critical periods" in cortical development when neural circuits undergo large adaptations in response to the environment (see review by Takesian and Hensch, 2013). Interestingly, there is growing evidence that PV+ cells in adult hippocampal circuits are modulated by similar mechanisms during learning.

This premise is supported by the finding that the trophic factor neuregulin1 (NRG1), which is a critical element in PV+ maturation during development, acts on adult PV+ cells to increase GABA transmission, resulting in a suppression of LTP induction at SC-CA1 synapses (Pitcher et al., 2008; Chen et al., 2010). When the NRG1 receptor, ErbB4, was selectively knocked out from PV+ cells, LTP at SC-CA1 synapses was increased and no longer repressed by NRG1. Interestingly, the PV+ cell specific ErbB4 knockout animals display a deficit in contextual fear conditioning, revealing an important role of PV+ cells in hippocampal learning (Chen et al., 2010). A recent and compelling study by Donato et al. (2013) has shown that PV+ basket cells in area CA3 show a change in activity state following contextual fear conditioning or environmental enrichment, two treatments found to respectively decrease or improve performance of hippocampal-dependent novel object recognition. The intensity of PV staining at axonal terminals was used as an indicator of PV+ cell activity state: high PV levels in the non-plastic state and low PV levels in the highly-plastic state. These observations are consistent with previous reports that pre-synaptic PV levels are able to modulate pre-synaptic calcium levels and GABA release in cerebellar interneurons during development (Collin et al., 2005). Furthermore, manipulations of the perineuronal net, an extracellular matrix that grows and shrinks during developmental "critical periods" and releases NRG1 and other PV+ cell modulators, can reset the PV+ cells to a highly-plastic low PV condition in the adult hippocampus (Donato et al., 2013) indicating that additional factors regulating PV+ cell excitability during development may control plasticity in the adult.

#### **PERSPECTIVE**

At the circuit level, a recent study reported that cholinergicallydriven θ-oscillations in CA1 involves an inhibitory circuit consisting mainly of CCK+ interneurons (Nagode et al., 2014). This conclusion was based on the observation that cholinergicallydriven oscillatory IPSCs were sensitive to cannabinoids and optogenetic silencing of CCK+ cells, but not PV+ cells. Thus, while CCK+ cells may be involved in low frequency oscillations such as θ-rhythm, PV+ cells may play a more prominent role in faster rhythms such as γ and sharp wave ripple oscillations. However, removing inhibition onto PV+ interneurons also affects θ-oscillation *in vivo* in CA1, suggesting a complex interaction between different interneuron types in oscillatory activity (Wulff et al., 2009). With the high diversity of PV+ interneurons in the hippocampus, there is also a very large level of diversity of synaptic plasticity. Recordings performed *in vivo* that take into account the axonal arbors and cell-type specific markers of interneurons, have revealed that each cell type has a potential role in the network activity of the hippocampus during a specific behavior (for example, Klausberger et al., 2003, 2004; Tukker et al., 2007; Lapray et al., 2012). Distinct differences in PV+ cell projection patterns and activities have been found in different hippocampal regions (Tukker et al., 2013). Even with all of this complexity, *in vivo* studies in which transmission from all PV+ cells has been removed reveal very interesting changes in behavior and hippocampal network activity (Korotkova et al., 2010; Murray et al., 2011; Royer et al., 2012). Deciphering how each subclass of interneuron dynamically contributes to network function during learning and disease states is a worthy goal for future work.

#### **REFERENCES**

the hippocampus. *J. Neurosci.* 33, 14567–14578. doi: 10.1523/jneurosci.0649-13. 2013

**Conflict of Interest Statement**: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Received: 31 January 2014; accepted: 13 March 2014; published online: 28 March 2014. Citation: Chevaleyre V and Piskorowski R (2014) Modulating excitation through plasticity at inhibitory synapses. Front. Cell. Neurosci. 8:93. doi: 10.3389/fncel.2014.00093 This article was submitted to the journal Frontiers in Cellular Neuroscience.*

*Copyright © 2014 Chevaleyre and Piskorowski. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# ADVANTAGES OF PUBLISHING IN FRONTIERS

FAST PUBLICATION Average 90 days from submission to publication

COLLABORATIVE PEER-REVIEW

Designed to be rigorous – yet also collaborative, fair and constructive

RESEARCH NETWORK Our network increases readership for your article

### OPEN ACCESS

Articles are free to read, for greatest visibility

#### TRANSPARENT

Editors and reviewers acknowledged by name on published articles

GLOBAL SPREAD Six million monthly page views worldwide

COPYRIGHT TO AUTHORS

No limit to article distribution and re-use

IMPACT METRICS Advanced metrics track your article's impact

SUPPORT By our Swiss-based editorial team

EPFL Innovation Park · Building I · 1015 Lausanne · Switzerland T +41 21 510 17 00 ·<EMAIL_ADDRESS>· frontiersin.org


**4. Keller–Dykhne Duality**

Circuit duality can be extended for a continuous system. As with circuits, the effective response of a continuous system is associated with that of the dual one; thus, a self-dual response is automatically guaranteed. Such a constraint can induce critical behaviors of self-dual systems. Here, these topics are reviewed. Next, we introduce differential forms to clearly extract the structure of the duality in a continuous system. The correspondence between the dualities in continuous and discrete systems is formulated through discretization of continuous fields. In this section, we always use the right-hand vector product, in other words, *A*, *B*, and *A* × *B* obey the right-hand rule.

#### *4.1. Two-Dimensional Resistive Sheets*

Consider an electric field *E*(*x*, *y*) and current density *K*(*x*, *y*) on a two-dimensional resistive sheet located at *z* = 0 with a sheet conductance *G*(*x*, *y*). Here, we assume that the fields inside the thin sheet are uniform along *z* and omit *z*-dependency for the fields. Thus, we treat the fields as two-dimensional vector fields independent of *z*. An example configuration is shown in Figure 24a. The physical dimensions of *E*, *K*, and *G* are V/m, A/m, and 1/Ω, respectively. From KVL and KCL, we obtain

$$
\nabla \times \mathbf{E} = \mathbf{0},
\tag{28}
$$

$$
\nabla \cdot \mathbf{K} = 0.\tag{29}
$$

**Figure 24.** (**a**) two-dimensional resistive sheet with boundary conditions; (**b**) solution for a sheet with a constant sheet conductance *G*(*x*, *y*) = *G*ref. Black lines represent the current flow. The potential is shown as a color map with isopotential gray contours.

Note that these equations can be directly obtained from Maxwell's equations; by omitting the time-derivative terms for steady states, we can obtain Equations (28) and (29) from Faraday's law and the law of charge conservation. Ohm's law is given by

$$\mathbf{K} = \mathbf{G} \mathbf{E},\tag{30}$$

where *G* is a conductance and generally a 2 × 2 matrix. For a metallic electrode, the boundary condition is written as

$$
\mathfrak{n} \times \mathcal{E} = 0,
$$

where *n* is the unit vector normal to the boundary. For an open boundary, the boundary condition is given by

$$
\mathbf{u} \cdot \mathbf{K} = 0.\tag{32}
$$

From Equation (28), the electric field is represented by

$$E = -\nabla \varphi \tag{33}$$

 with a potential *ϕ*(*x*, *y*). Combining Equation (33) with Equations (29) and (30), we obtain

$$
\nabla \cdot G \nabla \varphi = 0. \tag{34}
$$

The boundary of the *i*-th electrode is specified by the Dirichlet boundary condition

$$
\varphi = \varphi\_{\dot{\nu}} \tag{35}
$$

which is constant along the boundary. The open boundary is given by the Neumann boundary condition

$$
\mathfrak{n} \cdot \mathbb{G} \,\nabla \varphi = 0. \tag{36}
$$

For a simplified system with a constant scalar conductance, we have the Laplace equation

$$
\nabla^2 \varphi = 0\tag{37}
$$

from Equation (34). An example solution of the Laplace equation calculated by COMSOL Multiphysics-R is shown in Figure 24b.

#### *4.2. Duality in Laplace Equation*

In this subsection, we discuss duality in Laplace equations [76]. The solution of the Laplace equation is called a *harmonic function*. A harmonic function can be considered as a part of a holomorphic function. To see this fact, consider a holomorphic function *w*(*z*) = *u*(*z*) + j*v*(*z*) with *z* = *x* + j*y*, *u*(*z*) = Re[*w*(*z*)], and *v*(*z*) = Im[*w*(*z*)]. The holomorphism leads to the Cauchy–Riemann equations:

$$\frac{\partial \mu}{\partial x} = \frac{\partial v}{\partial y'}\tag{38}$$

$$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.\tag{39}$$

These equations can be expressed as

$$
\begin{bmatrix}
\frac{\partial \upsilon}{\partial x} \\
\frac{\partial \upsilon}{\partial y}
\end{bmatrix} = J \begin{bmatrix}
\frac{\partial \mu}{\partial x} \\
\frac{\partial \mu}{\partial y}
\end{bmatrix} \tag{40}
$$

with *J* = -0 −1 1 0 , which induces counterclockwise 90◦ rotation. As we differentiate Equations (38) and (39) along *x* and *y*, respectively, and combine the results, we obtain

$$
\nabla^2 u = 0,\tag{41}
$$

which states that *u* is harmonic. Similarly, *v* is also harmonic. Here, *v* is called a *harmonic conjugate* of *u*.

If *u* is given, how can we obtain its harmonic conjugate *v*? Focusing on *v*(*x*, *y*) = *v*(*x*0, *y*0) + path:(*x*0,*y*0)→(*x*,*y*)(∇*v*) · d*r* with Equation (40), we have

$$\upsilon(\mathbf{x}, y) = \upsilon(\mathbf{x}\_0, y\_0) + \int\_{\text{path:}(\mathbf{x}\_0, y\_0) \to (\mathbf{x}, y)} (f \nabla u) \cdot \mathbf{d} \mathbf{r}\_\prime \tag{42}$$

where *r* = [*x y*]T and (*x*0, *y*0) is a fixed point. We have assumed that the considered region is simply connected to define Equation (42). When we consider a small displacement Δ*r* along a line *v*(*x*, *y*) = const., we have (*J*∇*u*) · Δ*r* = 0, which leads to ∇*u* Δ*r*. Therefore, *u* and *v* constitute an orthogonal coordinate around the point ∇*u* = 0, as shown in Figure 25a. In this subsection, the operation of taking the harmonic conjugate is treated as a duality transformation.

**Figure 25.** (**a**) holomorphic function *w*(*z*) = *u*(*z*) + j*v*(*z*) defines an orthogonal coordinate around a point with d*w*/d*z* = 0; (**b**) harmonic potential *ϕ* and the lines of force −∇*ϕ*; (**c**) harmonic conjugate *ψ* for *ϕ* and the lines of force −∇*ψ*.

The above result induces duality for the potential problem of the Laplace equation. Let *ψ* be the harmonic conjugate of a harmonic potential *ϕ*. The relation between *ϕ* and *ψ* is depicted in Figure 25b,c. Now, we come back to resistive sheet problems. The current stream lines and isopotential contours are replaced with each other under the harmonic conjugate as shown in the simplest example of Figure 26. Furthermore, we can see that the harmonic conjugate interchanges the Dirichlet and the Neumann boundary conditions because of the 90◦ rotation.

**Figure 26.** Current and potential distributions for (**a**) original and (**b**) its dual resistive sheets with a uniform conductance.