Milan Klöwer
email: milan.kloewer@physics.ox.ac.uk

Malte F Jansen

Martin Claus

Richard J Greatbatch

Soeren Thomsen

Milan Kl

S Ören Thomsen

Sören Thomsen

Energy budget-based backscatter in a

Keywords: Backscatter, Energy budget, Negative viscosity, Eddy parameterization, Eddy-permitting, Mesoscale, Double gyre, Shallow water model

teaching and research institutions in France or abroad, or from public or private research centers.

Mesoscale eddies are a key turbulent feature of the global oceans [START_REF] Chelton | Global observations of 470 nonlinear mesoscale eddies[END_REF]. Understanding their complex interplay with the large-scale circulation [START_REF] Marshall | Eddy-mean flow interaction in a barotropic ocean model[END_REF]Wunsch & Ferrari, 2004) is a major challenge in climate research, inevitably important to assess and predict climate change (Randall 5 et al., 2007;[START_REF] Flato | 495 Evaluation of Climate Models. Climate Change 2013: The Physical Science Basis[END_REF]Vallis, 2016). Eddies are capable of transferring energy across scales and are therefore a crucial element in the ocean's energy cycle [START_REF] Rhines | Geostrophic Turbulence[END_REF][START_REF] Molemaker | Baroclinic Instability and Loss of Balance[END_REF][START_REF] Ferrari | Ocean Circulation Kinetic Energy: Reservoirs, Sources, and Sinks[END_REF][START_REF] Aiki | Energetics of the Global Ocean: the Role of Mesoscale Eddies[END_REF]. Finding a closure for eddies in ocean general circulation models is an active field of research [START_REF] A C C E P T E D M A N U S C R I P T Eden | Closing the energy cycle in an ocean model[END_REF][START_REF] Von Storch | An Estimate of the Lorenz Energy Cycle for the World Ocean Based on the STORM/NCEP Simulation[END_REF]Eden & Greatbatch, 10 2008). In addition to the energy cycle, eddies are also important for tracer transport and dominate the mixing and stirring of physical, chemical and biological water mass properties [START_REF] Gent | Parameterizing Eddy-Induced Tracer Transports in Ocean Circulation Models[END_REF]Abernathey et al., 2010;[START_REF] Liu | Adjoint-Based Estimation of Eddy-Induced Tracer Mixing Parameters in the Global Ocean[END_REF].

In the near future, state-of-the-art climate models will approach spatial res-15 olutions at which the largest eddies can be resolved explicitly [START_REF] Eyring | Overview of the Coupled Model Intercomparison Project Phase 6 (CMIP6) experimental design and organization[END_REF]. Despite the increasing performance of supercomputers, the most advanced Earth system models are not expected to fully resolve the mesoscale let alone the submesoscale [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF] within the next decades. Western boundary currents (e.g. the Gulf Stream and Kuroshio) and their exten-20 sion regions are an essential part in the wind-driven horizontal gyre circulation [START_REF] Imawaki | Ocean Circula-530 tion and Climate -A 21st Century Perspective[END_REF][START_REF] A C C E P T E D M A N U S C R I P T Maximenko | Ocean Circula-560 tion and Climate -A 21st Century Perspective[END_REF]. These regions show an enhanced eddy activity on scales of several hundreds of kilometres down to the subme-

A C C E P T E D M

A N U S C R I P T soscale [START_REF] Ferrari | The distribution of eddy kinetic and potential 490 energies in the global ocean[END_REF]. The need for more sophisticated turbulence closures, realistically parameterizing the e↵ect of meso-and submesoscale eddies 25 on the resolved flow, is hence clear.

Traditional approaches to eddy parameterization employ down-gradient fluxes [START_REF] Gent | Isopycnal Mixing in Ocean Circulation Models[END_REF], ensuring numerical stability [START_REF] Gri Es | Fundamentals of ocean climate models[END_REF], although evidence exists that eddies also a↵ect the mean circulation via upgradient momentum fluxes [START_REF] Greatbatch | Transport driven by eddy momentum fluxes in the Gulf Stream Extension region[END_REF]Wang et al., 2017). Some 30 approaches introduce stochasticity to account for sub-grid scale variability and parameterizations are formulated as a function of the resolved flow and random numbers, aiming to increase the number of variables that determine the future evolution of the flow field [START_REF] Palmer | A nonlinear dynamical perspective on model error: A 580 proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models[END_REF][START_REF] Berlo↵ | Random-forcing model of the mesoscale oceanic eddies[END_REF][START_REF] Jansen | Parameterizing subgrid-scale eddy effects using energetically consistent backscatter[END_REF][START_REF] Andrejczuk | Oceanic Stochastic Parameterizations in a Seasonal Forecast System[END_REF]Zanna et al., 2017). Physically motivated is the imple-35 mentation of eddy viscosity as a di↵usive turbulence closure to mimic the general tendency of the eddy field to mix and hence smooth gradients [START_REF] Smagorinsky | General Circulation Experiments With the Primitive Equations[END_REF][START_REF] Leith | Di↵usion Approximation to Inertial Energy Transfer in Isotropic Turbulence[END_REF][START_REF] Redi | Oceanic Isopycnal Mixing by Coordinate Rotation[END_REF][START_REF] Gri Es | Biharmonic Friction with a Smagorinsky-Like Viscosity for Use in Large-Scale Eddy-Permitting Ocean Models[END_REF]. Other approaches aim to parameterize the advective e↵ects of eddies in baroclinic instability and mixed layer stratification [START_REF] Gent | Isopycnal Mixing in Ocean Circulation Models[END_REF]Fox-Kemper & Ferrari, 40 2008;[START_REF] Fox-Kemper | Parameterization of mixed layer eddies. III: Implementation and impact in global ocean climate simulations[END_REF][START_REF] Brüggemann | Evaluating Di↵erent Parameterizations 460 for Mixed Layer Eddy Fluxes induced by Baroclinic Instability[END_REF], but do not include considerations of the kinetic energy cycle. Generally speaking, there are two main requirements that an eddy closure should satisfy in ocean circulation models: (i) the physical parameterization of instabilities, their stirring and mixing, and subsequent dissipation; and (ii) numerical stability through dissipation 45 of enstrophy at the grid scale. The viscosity in general circulation models is usually tuned to satisfy (ii), whereby viscosity coe cients are several orders of magnitude larger than would result from molecular considerations [START_REF] Gri Es | Biharmonic Friction with a Smagorinsky-Like Viscosity for Use in Large-Scale Eddy-Permitting Ocean Models[END_REF]. This leads to a significant dissipation of energy in most approaches. We therefore believe successful parameterizations should aim to close 50 the energy cycle by reducing the e↵ective viscosity that otherwise leads to a spurious energy dissipation at the grid scale [START_REF] A C C E P T E D M A N U S C R I P T Eden | Closing the energy cycle in an ocean model[END_REF][START_REF] Jansen | Parameterizing subgrid-scale eddy effects using energetically consistent backscatter[END_REF]Arbic et al., 2007). 2017)). A recently proposed approach [START_REF] Jansen | Parameterizing subgrid-scale eddy effects using energetically consistent backscatter[END_REF][START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF] involves the combination of a hyperviscous closure in order to remove energy and enstrophy from the grid scale with a prognostic variable for the sub-grid eddy kinetic energy (EKE). The dissipated EKE is conserved and 60 progressively re-injected into the resolved flow at larger scales via a deterministic term that is formulated as negative Laplacian viscosity. The e↵ectively reduced dissipation allows for higher levels of EKE, thus hopefully improving the mean state and large-scale variability. The approach was shown to be successful in an idealised ocean model in a channel configuration [START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF]. Here, 65 we apply and extend the parameterization to a shallow water model driven by double gyre wind forcing with an idealized basin geometry and no-slip boundary conditions, and provide evidence for its general applicability.

This study is structured as follows. Section 2 briefly presents the shallow water model used throughout this study and its energetics. The formulation 70 of sub-grid scale EKE and the resulting energy budget-based backscatter parameterization is introduced subsequently. In section 3.1 we discuss the model bias of the low resolution control run without parameterization with respect to the high resolution truth as reference. The impact of the energy budget-based backscatter parameterization on the low resolution model is analysed in section 75 3.2, and discussed in section 4. Finally, we summarise the results in section 5.

Methods

The shallow water model

The shallow water model [START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF][START_REF] Vallis | Atmospheric and Ocean Fluid Dynamics[END_REF] of the prognostic variables velocity u = (u, v) and sea surface elevation ⌘ that is used in this study is at the northern boundary. b z is a unit vector pointing in the upwards vertical direction. Boundary conditions are no-slip, as free-slip conditions were found to lead to a highly unrealistic representation of the western boundary current.

80 @u @t + (u • r)u + fb z ⇥ u = gr⌘ + F + B + D + ⌅ (1a) @⌘ @t + r • (uh) = 0. ( 1b

90

Steady and solely zonal double gyre wind forcing F = (F x , 0)

F x = ⌧ ⇢h , ⌧ = F 0 cos(2⇡yL 1 ⇡) + 2 sin(2⇡yL 1 ⇡) (2)
with amplitude F 0 = 0.12 Pa is used (Fig. 1d). Model simulations start from rest (u = 0) and the sea surface elevation is undisturbed (⌘ = 0) in the initial conditions. Bottom friction B is represented as a quadratic drag B = cD h |u|u with dimensionless drag coe cient c D = 10 5 , which reduces energy primarily

D = ⌫ B h 1 r • (hS(h 1 r • hS)), S(u, v) = 0 @ u x v y v x + u y v x + u y (u x v y ) 1 A (3)
with a constant viscosity coe cient ⌫ B , that depends on the grid spacing x as ⌫ B = 540 m 2 s 1 x 3 30km . The resulting Munk boundary layer [START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF] is resolved with at least one grid cell for x  30 km. The symmetric stress tensor

A C C E P T E D M A N U S C R I P T di↵usion.
We call the biharmonic stress tensor S ⇤ = S(h 1 r • hS(u, v)). The term ⌅ represents backscatter from unresolved scales and will be defined later.

For the unparameterized control runs we set ⌅ = 0. The advection of momentum is split into a vorticity part and a gradient of kinetic energy, and we use the 110 energy and enstrophy conserving scheme proposed by [START_REF] Arakawa | A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations[END_REF] to discretise the former. Similar to the arguments of [START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF], we favour this scheme as a sophisticated energy and enstrophy budget is central to the study. Without backscatter, the energy budget of the shallow water equations is

115 @ @t h 1 2 ⇢h|u| 2 + 1 2 g⇢⌘ 2 i = hu⌧ i h⇢c D |u| 3 i h⇢⌫ B hru • S ⇤ i (4)
where the first term on the left-hand side represents kinetic energy (KE) and the second potential energy (PE). The brackets denote a horizontal domainwide integral. These terms are further Reynolds-decomposed into mean and eddy contributions to kinetic and potential energy (MKE, EKE, MPE, EPE), respectively, The sea surface elevation is decomposed into a time mean and anomalies as

120 MKE = 1 2 ⇢h(b u 2 + b v 2 ), EKE = 1 2 ⇢h(u 00 2 + v 00 2 ), MPE = 1 2 g⇢⌘ 2 , EPE = 1 2 g⇢⌘ 02 . (5) A C C E P T E D M A N U S C R I P T
⌘ = ⌘ + ⌘ 0 .
Velocities are time averaged with thickness-weighting b u = hu/h and similarly for v [START_REF] Aiki | Energetics of the Global Ocean: the Role of Mesoscale Eddies[END_REF]. The respective anomaly is u 00 = u b u. In Eq. (4), the term hu⌧ i represents power input due to wind forcing. The second term on the right-hand side denotes dissipation due to bottom friction 125 and the third term on the right-hand side represents KE dissipation due to the biharmonic viscosity. Although harmonic di↵usion is always down-gradient, biharmonic di↵usion operators are not sign-definite and can also lead to power input [START_REF] Gri Es | Fundamentals of ocean climate models[END_REF] but are mostly an energy sink, as will be discussed later. Omitting the constant density ⇢, the EKE transfer to the unresolved flow associated with biharmonic dissipation follows from Eq. ( 4)

Formulation of the energy budget-based backscatter parameterization

150 Ėdiss = c diss ⌫ B hru • S ⇤ (7)
This term is negative where the biharmonic di↵usion removes energy from the resolved flow. In contrast to [START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF] we introduce a spatially and temporally varying scaling for c diss , which obeys 0  c diss  1, to allow only a fraction of the dissipated EKE to enter the sub-grid EKE budget. This is done for the following reasons: 155 (i) In the real ocean, the direction of energy cascades (upscale or downscale) depends on the local geostrophic balance and baroclinicity [START_REF] Scott | Direct Evidence of an Oceanic Inverse Kinetic Energy Cascade from Satellite Altimetry[END_REF]Arbic et al., 2007;[START_REF] Capet | Mesoscale to Submesoscale Transition in the California Current System. Part I: Flow Structure, Eddy Flux, and Observational Tests[END_REF][START_REF] Ferrari | Ocean Circulation Kinetic Energy: Reservoirs, Sources, and Sinks[END_REF][START_REF] Molemaker | Balanced and un-565 balanced routes to dissipation in an equilibrated Eady flow[END_REF][START_REF] Brüggemann | Routes to Dissipation under Di↵erent Dynamical Conditions[END_REF]. A varying c diss allows for an EKE transfer to the sub-grid EKE budget that is conditioned on prop- The loss of energy via forward cascades arising in unbalanced flows is represented using

c diss = ✓ 1 + R o R diss ◆ 1 (8)
with the Rossby number R o = |D| f estimated locally via the deformation rate [START_REF] Rhines | Geostrophic Turbulence[END_REF][START_REF] Greatbatch | Transport driven by eddy momentum fluxes in the Gulf Stream Extension region[END_REF][START_REF] Molemaker | Balanced and un-565 balanced routes to dissipation in an equilibrated Eady flow[END_REF][START_REF] Nadiga | Nonlinear evolution of a baroclinic wave and imbalanced dissipation[END_REF]. This is motivated by the fact that geostrophic turbulence at small Rossby numbers tends to undergo an upscale cascade of energy [START_REF] Scott | Direct Evidence of an Oceanic Inverse Kinetic Energy Cascade from Satellite Altimetry[END_REF], whereas large Rossby numbers are indicative of a downscale cascade where energy should in-185 deed be dissipated [START_REF] Brüggemann | Routes to Dissipation under Di↵erent Dynamical Conditions[END_REF][START_REF] Molemaker | Balanced and un-565 balanced routes to dissipation in an equilibrated Eady flow[END_REF][START_REF] Ferrari | Ocean Circulation Kinetic Energy: Reservoirs, Sources, and Sinks[END_REF]. In that sense, we interpret R diss as a cut-o↵ Rossby number beyond which loss of balance can lead to a strong forward energy cascade (Fig. 2b)

170 |D| = p (u x v y ) 2 + (u y + v x ) 2 . Eq. ( 8
The backscatter forcing term in the shallow water equations is formulated

190 A C C E P T E D M A N U S C R I P T
similar to the biharmonic viscosity, but in the corresponding harmonic form with a negative viscosity coe cient to allow for an upscale transfer of energy.

⌅ = h 1 r • ⌫ back hS, ⌫ back = c back x r max(2 e h , 0). ( 9
)
Using a shallow water model, we consider the vertically integrated sub-grid EKE with units m 3 s 2 , which alters the scaling for ⌫ back slightly through dividing by h, in contrast to the original formulation in [START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF], so that ⌫ back 195 retains its physical units as m 2 s 1 . c back is an O(1) non-dimensional constant, which we set as originally proposed to be 0.4 and do not perform sensitivity experiments as the dependence on this parameter was previously shown to be weak. Eq. ( 6) can predict negative values of e, hence using the maximum guarantees no backscatter in this case.

200

Similar to the derivation that led to Eq. ( 7), the backscatter term decreases the sub-grid EKE as

Ėback = ⌫ back hru • S. (10)
and transfers that energy back to the resolved flow via negative viscosity. A list of all model runs used in this study is given in Table 1. described in Table 1. The spin-up phase of 5 years is omitted for all analyses. Kinetic energy is defined in Eq. ( 4).

List of model runs

N x [km] ⌫ B [m 4 s 1 ]

Results

extensive and carry more transport at high resolution. These counter-rotating gyres share similarities with those described by [START_REF] Greatbatch | Four-Gyre Circulation in a Barotropic Model with Double-Gyre Wind Forcing[END_REF] as potential vorticity gradients are flattened towards the boundary, but in a much weaker sense.

225

Comparing the energy levels between the high and low resolution model, we conclude that the low resolution model lacks kinetic energy in the spatial and climatological mean by a factor of 2 to 2.5 (Fig. 4). This is mainly due to missing EKE at low resolution. The reservoir of EKE in the central western part of the domain, which is a factor of 2 larger at high resolution, dominates 230 the energy budget (Fig. 5). The MKE is only large in the western boundary current and less dependent on resolution. This supports the notion that only the boundary current of the basin-wide circulation is a persistent strong current, whereas the instantaneous flow in other parts of the domain is dominated by eddies. Potential energy is one order of magnitude lower than kinetic energy, but 235 similar conclusions hold concerning resolution-dependence (Fig. 5), as kinetic and potential energy are directly related in quasi-geostrophic flow [START_REF] Vallis | Atmospheric and Ocean Fluid Dynamics[END_REF].

Although the biharmonic viscosity is known to remove energy at the grid scale [START_REF] Gri Es | Biharmonic Friction with a Smagorinsky-Like Viscosity for Use in Large-Scale Eddy-Permitting Ocean Models[END_REF][START_REF] Jansen | Parameterizing subgrid-scale eddy effects using energetically consistent backscatter[END_REF][START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF], it a↵ects all spatial scales as seen in the EKE spectrum (Fig. 6). Most EKE is

245

In order to understand the di↵erences in eddy energy that come with di↵erent resolution, the sources and sinks to the energy budget are analysed. Wind forcing is found to be locally either an energy source or an energy sink, which depends on the direction of the zonal flow compared to the direction of the wind forcing (Fig. 7a andg), but less so on the resolution. In a spatially integrated physical settings of this study, and approximately independent of the resolution.

Bottom friction removes energy primarily from the western boundary current and its extension, where flow speeds are fastest (Fig. 7b andh). The pattern therefore resembles that of MKE (Fig. 5a andi). The power due to the viscosity 255 term reveals an energy sink in the eddy-dominated central western part of the domain, which is strong at low resolution but much weaker at high resolution (Fig. 7c andi). The viscosity in connection with no-slip boundary conditions also removes a substantial part of the energy from the boundary. Due to a not sign-definite biharmonic viscosity operator, some of the energy that is removed 260 directly at the boundary is partly re-injected a few grid cells further inward. At high resolution 64% of dissipation is through bottom friction, and only 36% due Distributions of Rossby numbers reveal that the low resolution model is not 265 able to reproduce the turbulent nature of the simulated flow at high resolution as Rossby numbers are in general lower by a factor of 2 (Fig. 2a). Based on this and the other diagnostics shown (Fig. 1,4567), we conclude that a low resolution model di↵ers statistically from its high resolution version that is taken as reference. For lower resolution, it is necessary to increase the viscosity coe cient in 270 the model (Table 1) to prevent numerical instabilities. This pushes the dissipative scales into the eddy scales, leading to a depression of eddy activity [START_REF] Nadiga | Alternating zonal jets and energy fluxes in barotropic wind-driven gyres[END_REF]. The unrealistically high viscosity therefore a↵ects the physical regime of the flow, measured here in terms of the Rossby number. Decreased eddy kinetic and potential energy follow, a↵ecting the mean circulation and the 275 variability of the flow.

Energy-budget based backscatter

The spurious viscous energy dissipation in the low resolution model can be counteracted using the backscatter approach. In the original formulation by [START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF] all dissipated EKE is transferred to the sub-grid EKE 280 budget (i.e. c diss = 1 in Eq. 7). The case of c diss = 1 is not going to be discussed here, as significant spurious energy tendencies occur due to excessive numerical noise at the boundaries. Instead, our strong backscatter simulation, which closely resembles this limit, uses a dissipation weakly dependent on the Rossby number, with R diss = 64 (see Eq. 8).

285

Having no-slip boundary conditions, a large amount of energy is dissipated via viscosity due to strong shear flows at the boundary (Fig. 7c andi), especially at low resolution. Consequently, most sub-grid EKE is concentrated in the western boundary region in the climatological mean (Fig. 8c). Additionally, the central western interior shows increased sub-grid EKE levels as following 290 from a pronounced EKE dissipation via viscosity in the highly turbulent region between the sub-tropical and sub-polar gyre (Fig. 7c). The explicit di↵usion

A C C E P T E D M A N U S C R I P T
of sub-grid EKE in its prognostic equation has a negligible e↵ect as the time scale for the grid-scale di↵usion is on the order of 20 days for ⌫ e = 540 m 2 s 1 , whereas the backscatter time scale, defined as the residence time of sub-grid 295 EKE, is on the order of hours to a day. The negligible role of sub-grid di↵usion is also supported by an experiment with an increased sub-grid EKE di↵usion (⌫ e = 5400 m 2 s 1 ), which reproduces virtually the same results. Solely negative values of sub-grid EKE are absent in the climatological mean (Fig. 8) due to the increased di↵usion, but with a negligible impact on the resolved flow. negative sub-grid EKE as the biharmonic dissipation is not sign-definite but are not accounted for in the backscatter term (Eq. 9).

The source of energy to the resolved flow via the backscatter parameterization increases with the local level of sub-grid EKE (Eq. 9). It follows therefore that the backscatter parameterization leads to a particularly strong forcing near the boundaries. In the strong backscatter case, the EKE spectrum shows a strong energy increase at the grid scale, indicative of numerical noise (Fig. The only considered parameter to tune the flow-regime dependent backscatter strength is the positive scalar R diss . We find for R diss < 1, the e↵ect of the 315 backscatter to be negligible, such that the low resolution run with backscatter converges towards its unparameterized control run. A moderate backscatter parameterization (R diss = 8) is tuned so as to match the kinetic energy level of the high resolution run (Fig. 4). Remarkably, this also coincides with a corrected EKE spectrum (Fig. 6).

320

The climatological mean of EKE (Fig. 5) in the moderate backscatter case is virtually identical to its reference at high resolution. Also EPE is considerably improved and relative contributions of the di↵erent energy reservoirs to the total energy are in good agreement with those of the high resolution run (Fig. 5). The energy cycle is corrected, as bottom friction accounts for most dissipation (Fig. 325 7e). Combining the energy sources and sinks of the biharmonic viscosity and the backscatter term reveals the net e↵ect of the sub-grid parameterization (Fig. 7f), which is in overall agreement with viscous dissipation at high resolution (Fig. 7i). Notice that the patterns in Figs. 7f and7i are not expected to be identical, as the two represent energy exchange between the resolved and sub-330 grid flow at di↵erent cut-o↵ scales. Importantly, however, the total sub-grid energy dissipation rate, which represents the magnitude of the forward energy cascade, is similar in the two simulations.

Increased EKE levels coincide with a higher eddy activity, especially in regions where eddies are absent without backscatter parameterization (Fig. 3).

335

This has a positive e↵ect on the climatological mean circulation as the eddydriven circulations along the northern and southern boundaries are in much better agreement with those simulated at high resolution (Fig. 1). Furthermore, a moderate strength backscatter parameterization yields Rossby numbers distributions that closely resemble those simulated at high resolution (Fig. 2a). maker et al., 2010;[START_REF] Ferrari | Ocean Circulation Kinetic Energy: Reservoirs, Sources, and Sinks[END_REF][START_REF] Rhines | Geostrophic Turbulence[END_REF]. We are thus able to eliminate the numerical distortions at the western boundary that would otherwise limit the practicality of the backscatter parameterization. The eddy-mean flow interaction in a low resolution model is shown to be enhanced, enabling 350 the model to simulate an improved climatological mean state, that presumably results from increased levels of EKE and EPE due to a more sophisticated simulation of the energy cycle. We want to emphasise that the tuning of the backscatter strength is solely motivated by matching reference kinetic energy levels. Remarkably, main aspects of the mean circulation are improved without 355 tuning towards an optimised mean state.

For the sake of simplicity, computational power and given the importance of horizontal scales, we have focussed here on the two dimensional shallow water equations, which we find adequate to simulate eddy-mean flow interactions as a predominant process in the spectrum of geostrophic turbulence in the global 360 oceans. An idealised ocean basin is considered, which represents a sub-tropical to sub-polar ocean basin as a rectangular domain without realistic coast lines or bottom topography. Nevertheless, we suggest that the simplified geometry does not influence the major results of this study because processes of largescale geostrophic turbulence occur similarly in all ocean basins with variable 365 coastlines and bottom topography [START_REF] Chelton | Global observations of 470 nonlinear mesoscale eddies[END_REF][START_REF] Von Storch | An Estimate of the Lorenz Energy Cycle for the World Ocean Based on the STORM/NCEP Simulation[END_REF].

The shallow water model considered does, however, have several shortcomings that limit the comparability to the real ocean. Many of the processes that lead to a forward cascade of energy in the real ocean (such as interactions with bottom topography and internal waves) are missing in our model (e.g. Gertz For simplicity, we used a constant viscosity coe cient for the biharmonic lat-375 eral mixing of momentum. More sophisticated approaches exist [START_REF] Smagorinsky | General Circulation Experiments With the Primitive Equations[END_REF][START_REF] Leith | Di↵usion Approximation to Inertial Energy Transfer in Isotropic Turbulence[END_REF][START_REF] Gri Es | Biharmonic Friction with a Smagorinsky-Like Viscosity for Use in Large-Scale Eddy-Permitting Ocean Models[END_REF], promoting the use of a flow-and scale-aware non-linear viscosity coe cient in order to remove enstrophy from the grid-scale and satisfy numerical stability. However, none of these approaches simultaneously dissipate enstrophy without also dissipating excessive amounts 380 of energy on the grid scale, the primary motivation for using the backscatter approach. A Smagorinsky-like formulation of the biharmonic viscosity was successfully combined with the backscatter parameterization [START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF].

As the constant viscosity coe cient used here provides comparable improvements, we believe that the energy-budget based backscatter parameterization 385 is applicable to various forms of di↵usion operators as long as viscosity is in a biharmonic form (or of even higher order) and backscatter is formulated with a harmonic operator.

Summary and conclusion

An energy budget-based backscatter parameterization for the unresolved 390 scales in a shallow water model driven by double gyre wind forcing is presented.

The parameterization is an extension of the original approach by Jansen et al.

(2015b) using a biharmonic viscosity in combination with an additional deterministic forcing term, formulated as negative Laplacian viscosity, to represent backscatter of energy from unresolved scales [START_REF] Jansen | Parameterizing subgrid-scale eddy effects using energetically consistent backscatter[END_REF]. The ki-395 netic energy that is removed from the resolved flow enters the sub-grid EKE budget, which is introduced as a prognostic variable. Based on local levels of sub-grid EKE, the previously dissipated energy is re-injected at larger spatial scales via the backscatter term, corresponding to an artificial upscale transfer

•

  Generalization of the energy budget backscatter for a sub-grid eddy parameterization • Rossby number-dependent dissipation scaling controls the forward energycascade • Parameterization is tested successfully in a double gyre basin shallow water model • The energy cycle of the model is considerably improved at low computa-

  Eddy parameterizations that allow for upscale and downscale fluxes of en-A C C E P T E D M A N U S C R I P T ergy are under investigation (e.g. Berlo↵ (2005); Nadiga (2008); Eden (2010); 55 Marshall et al. (2012); Porta Mana & Zanna (2014); Mak et al. (2016); Zanna et al. (

  We use a similar barotropic (gravity g = 10 ms 1 , density ⇢ = 10 3 kgm 3 ) double gyre basin as in[START_REF] Cooper | Optimisation of an idealised ocean model, stochastic parameterisation of sub-grid eddies[END_REF]: To resemble a highly idealised North Atlantic, we use a square domain of size L 2 = 3840 km ⇥ 3840 km with Cartesian coordinates (x, y) centred at 30°N and a constant undisturbed ocean depth of H = 500 m. The layer thickness is h = ⌘ + H. With the 85 beta-plane approximation, the Coriolis parameter increases linearly from f = 3.5 ⇥ 10 5 s 1 at the southern boundary of the domain to f = 1.1 ⇥ 10 4 s 1

95

  on large scales by approximately 30% compared to simulations without bottom friction. The drag is two orders of magnitude weaker than proposed by Arbic &Scott (2008) as the flow speed at the bottom of the ocean is much smaller than the barotropic flow speed simulated by the model. In practice, a weak drag allows to maintain vigorous turbulence. Lateral mixing of momentum D 100 is represented as a biharmonic version of the di↵usion operator formulated by[START_REF] Shchepetkin | A Physically Consistent For-600 mulation of Lateral Friction in Shallow-Water Equation Ocean Models[END_REF]

Figure 1 :

 1 Figure 1: (a,b,c) Climatological mean circulation represented as stream lines calculated from (u, v). Colour and thickness of the stream lines are associated with speed. (d) Steady and solely zonal double gyre wind stress ⌧ .

Figure 2 :

 2 Figure 2: (a) Rossby number histogram for all grid points and time steps disregarding spinup. The histogram from the high resolution run is scaled to match the histograms at low resolution. Time and spatial mean Rossby numbers are marked as vertical ticks for all model runs in corresponding line styles and colours. Bins are evenly spaced on a logarithmic axis. (b) Rossby number dissipation scaling c diss for di↵erent backscatter strength controlled by R diss .

130A

  Following the ideas of[START_REF] Eden | Towards a mesoscale eddy closure[END_REF] for a mesoscale eddy closure, which are further developed in[START_REF] Jansen | Parameterizing subgrid-scale eddy effects using energetically consistent backscatter[END_REF] andJansen et al. (2015a), we seek to find an energy equation for the unresolved scales applied to the shallow water equations. Similar to[START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF] we complement the shallow water equations with the following prognostic equation for the 135 from the resolved flow due to biharmonic di↵usion is represented with Ėdiss . Adding this term to the prognostic equation of sub-grid EKE is therefore a transfer of energy from the resolved to the unresolved flow.The term Ėback represents the tendency associated with the backscatter, hence 140 the energy transfer from the sub-grid EKE budget back onto the resolved flow, which is realised with negative Laplacian viscosity. From the perspective of the sub-grid EKE, the first term on the right-hand side acts as a forcing, the second is a damping term, the third is a di↵usion term. We set a constant di↵usivity coe cient ⌫ e = 540m 2 s 1 and re = 0 serves as boundary condition. As in 145[START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF] we neglect any advection of sub-grid EKE with the mean flow due to a lack of an adequate theory that represents the combined e↵ect of advection and wave propagation.

  160 erties of the locally resolved flow. A C C E P T E D M A N U S C R I P T (ii) Having c diss < 1 yields an energy sink for the resolved flow, as the fraction 1 c diss is not subject to the recycling process of the backscatter parameterization. This is desired for reasons of numerical stability: Choosing c diss = 1, which corresponds to the original choice in Jansen et al. (2015b), 165 may lead to an e↵ective dissipation that is too weak, especially when other routes to dissipation are negligible.

  ) obeys the condition 0  c diss  1 for a positive non-dimensional constant R diss , which controls the backscatter strength and remains subject to tuning. Having a deformation rate-based scaling of dissipation via c diss means that regions of strong shear, especially boundary currents that are subject to no-slip boundary conditions, experience a stronger 175 dissipation than elsewhere, in agreement with Zhai et al. (2010). This avoids re-injecting locally an excess amount of energy via the backscatter term where it may lead to spurious oscillations at the grid scale, which could in turn cause numerical instabilities. Physically speaking, we want to overcome the spurious energy dissipation at the grid scale for balanced flows (R o ⌧ 1) but retain the 180 dissipative character for unbalanced flows with large Rossby number

Figure 3 :

 3 Figure 3: Snapshots of relative vorticity after 30 years of model integration. Positive vorticity is associated with anti-clockwise rotation and vice versa. Please note the non-linear colour map to highlight non-extreme structures.

2053. 1 .AFigure 4 :

 14 Figure 4: Spatially integrated kinetic energy for 30-year long integrations of model runs as

Figure 5 :

 5 Figure 5: Climatological mean of mean kinetic energy (MKE), eddy kinetic energy (EKE), mean potential energy (MPE) and eddy potential energy (EPE). Percent value in the lower right corner of each sub-plot denote the relative contribution to the total energy defined as the sum of MKE, EKE, MPE and EPE.

Figure 6 :

 6 Figure 6: Eddy kinetic energy spectrum. The spectral energy is logarithmically plotted against horizontal wavenumber K but for readability relabelled with the corresponding wavelength. A theoretical spectrum of K 3 is given for comparison. The barotropic Rossby radii of deformation for the southern (northern) boundary of the domain L max Ro (L min Ro ) are given for orientation.

Figure 7 :

 7 Figure 7: Climatological mean of energy sources and sinks due to wind forcing, bottom friction, viscosity and backscatter. Positive (negative) values indicate an energy source (sink). Backscatter power is zero for low and high resolution control run. In (c,f,i) the most extreme values occur at the boundary and are up to one order of magnitude beyond the color scale. The numbers in the top right of each sub-plot denote a domain-wide average in units of 10 3 Wm 2 .

  300

Figure 8 :

 8 Figure 8: Climatological mean of the sub-grid EKE variable e for di↵erent backscatter strength. Please note the non-linear colour map to highlight lower values. Grey areas have a

  305

  6). Snapshots, especially of relative vorticity, reveal that these numerical oscillations occur in the vicinity of boundary currents due to weak dissipation (not shown). Furthermore, in the strong backscatter case energy levels exceed those of the high resolution run on all scales (Fig.4 and 6), and the histogram ofA C C E P T E D M A N U S C R I P TRossby numbers (Fig.2a) reveals that high Rossby number flow is too prevalent, 310 whereas low Rossby number flow is too rare compared to the high resolution run. Hence, it would be desirable to decrease the strength of the backscatter parameterization, especially near the western boundary.

  While the original[START_REF] Jansen | Energy budgetbased backscatter in an eddy permitting primitive equation model[END_REF] parameterization is shown to lead to an excess of EKE at the grid scale in the vicinity of strong boundary currents with no-slip boundary conditions, the generalisation presented in this study overcomes this limitation by introducing a Rossby number-based dissipation 345 scaling that yields an energy sink for geostrophically unbalanced flow (Mole-

  2009)). Furthermore, only the vertically integrated sub-grid EKE is considered. It remains an open question how to parameterize vertical fluxes of sub-grid EKE in a more advanced three dimensional ocean general circulation model permitting baroclinic instability.

  S C R I P T of energy and hence a closure for the energy cycle. Simultaneously, enstrophy 400 is dissipated at the grid scale, ensuring numerical stability. A newly formulated Rossby number-based dissipation scaling allows for a route to energy dissipation for geostrophically unbalanced flow as supported by theory and observations. The backscatter parameterization presented in this study is shown to substantially improve the model, e↵ectively improving the simulation equivalent to 405 about a factor of two to four increase in resolution at negligible additional computational cost. While the implementation of an energy budget-based backscatter parameterization in a global climate model may not be straightforward, we provide evidence for its potential to substantially improve the models currently used for weather and climate prediction. We therefore highly promote further 410 research on energy budget-based backscatter methods.

Finally

  , we want to emphasise the importance of closing the energy cycle in an ocean circulation model[START_REF] A C C E P T E D M A N U S C R I P T Eden | Closing the energy cycle in an ocean model[END_REF] for successful climate simulations and a better prediction of climate change. Understanding the routes to dissipation in geophysical turbulence is a major challenge in theory and observations and is 415 an essential basis for sophisticated parameterizations of unresolved scales. The energy budget-based backscatter approach provides the potential to implement realistic routes to dissipation in an ocean circulation model, which provides a cornerstone for improved climate predictions. A C C E P T E D M A N U S C R I P T

Table 1 :

 1 List of model runs used in this study. The total number of grid cells is denoted with N , the grid spacing with x. ⌫ B is the biharmonic viscosity coe cient. The tuning parameter for the backscatter strength is R diss . The total cpu time tc is given in relation to the computing time of the low resolution control run. Please note that the total cpu time is only roughly estimated and heavily dependent on the computing architecture.

	R diss	t c

S is once evaluated with (u, v) and once with h 1 r • hS(u, v) for biharmonic

[START_REF] Greatbatch | Four-Gyre Circulation in a Barotropic Model with Double-Gyre Wind Forcing[END_REF][START_REF] Marshall | Eddy-mean flow interaction in a barotropic ocean model[END_REF]. The double gyre circulation is not obvious from snapshots, but the climatological mean velocities (Fig.1a and c) provide evidence for its existence. At low and high resolution a strong boundary current reaching speeds of about 1 ms 1 is simulated. The southern (northern) gyre is referred to as sub-tropical (sub-polar) gyre. The low resolution model shows 220 weak additional gyres in the north-east and south-east corner, which are more

Acknowledgements

Funding from the German Ministry for Education and Research (BMBF) through MiKlip2, subproject 01LP1517D (ATMOS-MODINI) is greatly acknowledged. RJG is also grateful to GEOMAR for continuing support. MFJ received

Conflict of interest 420

The authors declare that they have no conflict of interest.

Supplementary materials

Data relevant to this study is available in [START_REF] Klöwer | Data from shallow water model: Control simulations and backscatter experiments[END_REF]. A more detailed 435 description of the shallow water model used, its discretisation as well as model, analysis and plotting scripts can be found in [START_REF] Klöwer | milankl/swm: Shallow water model[END_REF].


Nadia Jaafari

Kaspar E Vogt

Peter Saggau

Loew M Leslie

Dejan Zecevic

Marco Canepari
email: marco.canepari@ujf-grenoble.fr

Combining Membrane Potential Imaging with Other Optical Techniques

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The advantages of the optical measurements of membrane potential (V m ), over electrode recordings, are the possibility to record voltage changes at high spatial resolution and from a very large number of sites and the ability to probe structures too small to tolerate electrodes. Combining this measurement with another optical measurement or with an optical stimulation at the same spatial resolution and scale allows investigating and understanding the voltage-dependence of a variety of physiological phenomena or the changes of V m produced by activation of a particular signaling pathway as well as by a well-defined stimulation protocol. In this chapter we describe in detail the main optical techniques that have been so far successfully combined with V m imaging: calcium (Ca 2+ ) imaging, photorelease of caged compounds and optogenetic photostimulation. At the end of this chapter we discuss the case of V m imaging combined with channelrhodopsin (ChR) stimulation, which is becoming a fundamental experimental approach in neurophysiology.

Combined optical measurements have been widely used in biology for a long time. In molecular biology, for instance, the concomitant localization of two or more proteins is achieved by using fluorescent antibodies with different absorption and emission spectra. Combined imaging of two different physiological parameters requiring sequences of signal acquisitions was more difficult to achieve until twenty years ago. Thus, before the introduction of more sophisticated imaging systems, combined optical recordings at relatively slow rate and using two indicators have been utilized to measure intracellular pH and Ca 2+ [START_REF] Martinez-Zaguilan | Simultaneous measurement of intracellular pH and Ca2+ using the fluorescence of SNARF-1 and fura-2[END_REF] as well as slow V m changes and Ca 2+ signals [START_REF] Kremer | Simultaneous fluorescence measurement of calcium and membrane potential responses to endothelin[END_REF]. Combined functional imaging is therefore an experimental approach that evolved in the last two decades in parallel with advancements in imaging technology. The correlation of a V m recording with a measurement of a Ca 2+ signal is perhaps the most important case of dual optical measurement of a functional signal (Canepari et al. 2008). Indeed, crucial information on Ca 2+ influx from the extracellular space comes from the fact that Ca 2+ signals are directly correlated with V m when they originate from Ca 2+ channels in the plasma membrane. This category includes voltage-gated Ca 2+ channels (VGCCs), glutamate receptors (in particular NMDA receptors) and other Ca 2+ permeable pores. In all these cases, the charge influx associated with Ca 2+ contributes, to a different extent, to the current underlying depolarization. In addition, the biophysical properties of several Ca 2+ signals in the plasma membrane also depend on membrane depolarization. For instance, Ca 2+ influx via VGCCs is voltage-dependent since depolarization is necessary for channel opening. Similarly, Ca 2+ influx via NMDA receptors is voltage-dependent since depolarization is necessary to unblock the channel from Mg 2+ . In contrast, Ca 2+ signals are not directly correlated with V m when they are due to Ca 2+ flux through internal membranes. For example Ca 2+ release from internal store is not directly correlated with V m . In this case, Ca 2+ flux does not contribute to the current underlying the V m change.

Among photostimulations combined to V m imaging, chemical stimulation delivered by photorelease of a molecule from a caged compound is a powerful tool to explore many physiological phenomena.

Notably, uncaging of principal neurotransmitter such as glutamate or GABA, allows investigating how local activation of synaptic receptors produces or modulates V m in different sub-cellular sites (Vogt et al. 2011b). V m imaging can be also combined with ChR stimulation expressed in sub-sets of well identified neurons. This application may provide new information on the function of local neuronal circuits [START_REF] Tsuda | Probing the function of neuronal populations: combining micromirror-based optogenetic photostimulation with voltage-sensitive dye imaging[END_REF].

The techniques of V m imaging described in this chapter are achieved by recording signals from organic voltage-sensitive dyes. Unless specified otherwise, this type of signal is expressed as a fractional change in fluorescence intensity ∆F/F 0 = (F(t) -F(t 0 ))/F(t 0 ), where F(t) is the recorded fluorescence intensity at a given time and F(t 0 ) is the fluorescence intensity at the resting membrane potential (resting fluorescence). Finally, most of the examples in this chapter are recordings from individually-stained single-neurons described in the previous chapter.

Principles of combining two optical techniques

The combination of two optical techniques is typically achieved by using two photosensitive molecules. These molecules can be indicators, for instance fluorescence dyes, or molecules that produce a stimulation of the cell upon light activation. If one of the two molecules is used for photostimulation, for instance a caged compound or a photo-sensitive protein, achieving simultaneous imaging requires that this molecule is not excited by the same wavelength used for excitation of the imaging indicator. This requirement is straightforward to achieve for many caged compounds that are activated exclusively by UV light and that can be combined with indicators excited in the visible range.

In contrast, combining imaging techniques with ChR stimulation is more difficult since ChR is activated by light over a broad visible spectrum. If the two molecules are both fluorescent indicators, for instance a voltage-sensitive dye and a Ca 2+ indicator, combined imaging can be achieved in three different ways depending on the optical properties of the two indicators. First, the two indicators may have separate absorption spectra but a largely overlapping region in the emission spectra. Second, the two indicators may have a large overlapping region in their absorption spectra but well separated emission spectra.

Third, the two indicators may have a separation both in the absorption and in the emission spectra. The optimization of recording signals from two indicators depends on the availability of probes with narrow excitation and emission bands. This is not the case for the available fast voltage sensitive dyes.

Most of the styryl voltage dyes are characterized by a broad excitation spectrum in the visible range and a broad emission spectrum in the red-IR region [START_REF] Fluhler | Spectra, membrane binding, and potentiometric responses of new charge shift probes[END_REF]. Fig. 4.2 shows the absorption and emission spectra of the hydrophobic voltage-sensitive dye di-8-ANEPPS which are similar to those of di-4-ANEPPS and of the water soluble indicators JPW-1114 (di-2-ANEPEQ) and JPW-3028 (di-1-ANEPEQ) all based on the same chromophore. Similar spectral properties are also shared by many of the commonly used "RH" dyes such as RH-795, RH-237, RH-421 and RH-414. With these dyes, well separated spectra are difficult to obtain using commercially available Ca 2+ indicators excited in the visible spectral range. Nevertheless, the combined recordings are still possible due to a large Stokes shift of charge-shift voltage probes (~150 nm). In addition, more efficient combined recordings can be achieved using Ca 2+ indicators excited in the UV spectral range.

Both voltage dyes and Ca 2+ indicators can be loaded into cells either by extracellular bath application of the indicator or by intracellular injection using either a patch-electrode or a sharp microelectrode. For extracellular loading, Ca 2+ dyes are commercially available in membrane-permeant AM-ester form. Once in the cell, the acetyl-ester is hydrolyzed by the endogenous esterases and the released free indicator remains in the cytoplasm [START_REF] Yuste | Loading Brain Slices with AM Esters of Calcium Indicators[END_REF]. For intracellular loading, Ca 2+ indicators are available as water-soluble potassium salts. When injected into cells from a patch electrode, the dye can quickly equilibrate in the cytosol at a given concentration permitting quantitative estimate of Ca 2+ signals [START_REF] Eilers | Dye Loading with Patch Pipettes[END_REF]. In general, intracellular loading of the Ca 2+ indicator is done in conjunction with the intracellular loading of the voltage sensitive dye. It is also possible to load the cell with the voltage sensitive dye first and include the Ca 2+ indicator in the pipette used for re-patching the labeled cell. In this protocol, V m imaging is carried out first, immediately after re-patching, during which time the cell is loaded with the Ca 2+ indicator from the recording pipette.

3 Combining V m and Ca 2+ imaging

Combining V m and Ca 2+ imaging in sequential way

Commonly used voltage sensitive dyes such as di-4-ANEPPS, di-8-ANEPPS and Di-2-ANEPEQ (JPW-1114) have wide excitation spectrum in the blue/green region and wide emission spectrum in the red/IR region (Fig. 4.2). If combined with Fluo, Calcium Green or Oregon Green Ca 2+ indicators, both dyes can be excited by a blue wavelength (which, however, is not optimal for the voltage-sensitive dye) and green (Ca 2+ ) and red (V m ) fluorescence can be recorded separately. Voltage sensitive dye excitation is optimal at 532 nm [START_REF] Canepari | Imaging Inhibitory Synaptic Potentials Using Voltage Sensitive Dyes[END_REF]. Thus, if combined with UV-excitable Fura indicators, the voltage-sensitive dye can be excited at its optimal visible wavelength and, as with the previous combination, the two emission signals can be recorded separately.

Although both types of combinations allow simultaneous measurement of V m and Ca 2+ fluorescence, many studies were initially performed through sequential recordings of the two optical signals.

Combined V m and Ca 2+ imaging from individual neurons using blue-excitable/green-emitting Ca 2+ indicators has been also used in several other studies. In the barrel cortex, the high-affinity indicator Oregon Green BAPTA-1 was combined with the voltage sensitive dye JPW-1114 in recordings from individual neurons and with the voltage indicator RH-1691 in in vitro and in vivo network recordings [START_REF] Berger | Combined voltage and calcium epifluorescence imaging in vitro and in vivo reveals subthreshold and suprathreshold dynamics of mouse barrel cortex[END_REF]. In prefrontal cortical neurons, dendritic recordings were obtained combining JPW-1114 either with the high-affinity Ca 2+ indicator Calcium Green-1 or with the low-affinity Ca 2+ indicator Fluo-5F [START_REF] Milojkovic | Voltage and calcium transients in basal dendrites of the rat prefrontal cortex[END_REF].

The use of UV-excitable Fura indicators minimizes the spectral overlap with the voltage-sensitive dye. Additionally, the voltage-sensitive dye is not excited during Ca 2+ -imaging preventing a substantial photodynamic damage that can occur during relatively long recording periods (Canepari et al., 2008).

UV-excitable Fura indicators with different equilibrium constants (K d ) are commercially available (see Table 4.1). This permits the choice of the most suitable indicator for the measurement of either the Ca 2+ influx (high-affinity indicator) or the change in intracellular free Ca 2+ concentration (low-affinity indicator). Invitrogen -Molecular Probes handbook (for fura-2 and bis-fura-2 at 1 mM Mg 2+ ); (b) Schneggenburger et al. 1999; (c) Hyrc et al. 2000; (d) [START_REF] Naraghi | T-jump study of calcium binding kinetics of calcium chelators[END_REF] In contrast to the indicators excited in the visible range, Fura indicators allow ratiometric measurements. When excited above their isosbestic wavelength (~360 nm), the changes of fluorescence associated with an increase in Ca 2+ concentration are negative. Imaging above the isosbestic wavelength has several advantages. First, the resting fluorescence, which corresponds to nominally 0 Ca 2+ , is significantly higher than with dyes that increase their fluorescence with Ca 2+ .

Second, the achievable imaging contrast is high because healthy cells are characterized by very low resting Ca 2+ , and the dye in adjacent structures, exposed to millimolar Ca 2+ concentrations, is essentially not fluorescent. Third, since the dynamic range of the fluorescence is ~1 after subtraction of fluorescence at saturating Ca 2+ , these measurements allow a better quantitative estimation of the Ca 2+ (Fig. 4.3b). Sequential V m and Ca 2+ recording using Fura Ca 2+ indicators has been done in CA1 hippocampal pyramidal neurons [START_REF] Canepari | Dendritic signals from rat hippocampal CA1 pyramidal neurons during coincident pre-and post-synaptic activity: a combined voltage-and calcium imaging study[END_REF], in prefrontal cortex layer-5 pyramidal neurons [START_REF] Milojkovic | Voltage and calcium transients in basal dendrites of the rat prefrontal cortex[END_REF]) and in cerebellar Purkinje neurons (Canepari and Vogt 2008). Sequential recording (as well as signal averaging) is meaningful only if repeated application of the same stimulation protocol results in the same response. This requirement must be confirmed experimentally by An important aspect of combined V m and Ca 2+ imaging is the interpretation of Ca 2+ optical signals, which depends on how much the Ca 2+ indicator perturbs the physiological Ca 2+ homeostasis. The buffering capacity of a Ca 2+ indicator, K dye , defined as the ratio between the dye-bound Ca 2+ and the free Ca 2+ in the presence of the indicator depends on the dissociation constant and on the concentration of the indicator (Table 4.1). The perturbation of the physiological Ca 2+ introduced by the Ca 2+ indicator can be evaluated by comparing the parameter K dye with the endogenous buffering capacity of the cell (K cell ). The interpretation of Ca 2+ optical signals is simplified when K dye is either much larger or much smaller than K cell , i.e. when most of Ca 2+ binds to the indicator or when the fraction of Ca 2+ bound to the indicator is negligible compared to the total Ca 2+ (Canepari et al. 2008). Without addressing in detail the issue of calibration of Ca 2+ optical signals [START_REF] Neher | Some quantitative aspects of calcium fluorimetry[END_REF] in this volume focused on V m imaging, we will note that, in the first case, the ∆F/F 0 signal associated with Ca 2+ increase is approximately linear with the total intracellular Ca 2+ signal. In the second case, the time course of the ratio (F-F min )/(F max -F), where F min and F max are the fluorescence intensities at 0 and saturating Ca 2+ respectively, is linear with the physiological intracellular free Ca 2+ concentration change (∆[Ca 2+ ] i ). In many instances, Ca 2+ signals are due to Ca 2+ influx through a Ca 2+ channel in the plasma membrane. The contribution to the change in membrane potential due to the Ca 2+ influx depends on the Ca 2+ permeability of the channel relative to its permeability to other ions, in particular to Na + and K + . This is different for VGCCs, AMPA receptors, NMDA receptors and other Ca 2+ permeable pores such as transient receptor potential (TRP) channels.

Vice-versa, because opening or unblocking of Ca 2+ channels often depends, in a non-linear manner, on the local membrane potential, the Ca 2+ influx is often, but not always, larger when the voltage related depolarizing optical signal is larger [START_REF] Canepari | Dendritic signals from rat hippocampal CA1 pyramidal neurons during coincident pre-and post-synaptic activity: a combined voltage-and calcium imaging study[END_REF]. Because this bi-directional relationship is complex, the analysis of V m and Ca 2+ signals from multiple sites on individual neurons is not straightforward and requires careful interpretation.

A notable application of combined V m and Ca 2+ imaging is the study of synaptic plasticity where the supra-linear Ca 2+ signal plays an important role. The supra-linear Ca 2+ signal occurs when two stimulation protocols are combined to evoke the coincident activity (pairing protocol) which results in a Ca 2+ signal that is larger than the sum of the two Ca 2+ signals associated with the application of individual stimulation protocols. A supra-linear Ca 2+ signal is always caused by a supra-linear ∆[Ca 2+ ] i , but not necessarily by a supra-linear Ca 2+ influx through the plasma membrane. Indeed, the supra-linear Ca 2+ signal might be caused by Ca 2+ release from internal stores or by the saturation of the endogenous Ca 2+ buffer. However, in the latter case, if the buffering capacity of the dye dominates over the endogenous buffering capacity of the cell and the dye is not saturated, the presence of the indicator will cancel the supra-linear ∆[Ca 2+ ] i and no Ca 2+ dependent ∆F/F 0 will be observed. Thus, in these conditions, detection of supra-linear Ca 2+ dependent ∆F/F 0 signals which are not due to Ca 2+ release from internal stores always corresponds to a supra-linear Ca 2+ influx and it must always correlate with an increase in the depolarizing V m signal. In contrast, if the buffering of the dye is negligible compared to the endogenous buffering of the cell, a supra-linear ∆[Ca 2+ ] i signal that is due to the saturation of the endogenous Ca 2+ buffer can be measured reliably. The two different types of supra-linear Ca 2+ signals not involving Ca 2+ release from stores are illustrated by the two following examples.

In measurements from CA1 hippocampal pyramidal neurons [START_REF] Canepari | Dendritic signals from rat hippocampal CA1 pyramidal neurons during coincident pre-and post-synaptic activity: a combined voltage-and calcium imaging study[END_REF]), cells were loaded with the voltage-sensitive dye JPW-3028 and 300 µM Bis-Fura-2. The purpose of these experiments was to analyze, over large regions of the dendritic tree, the V m and Ca 2+ optical signals associated with back-propagating action potentials, with excitatory postsynaptic potentials (EPSPs) and with pairing of these two membrane potential transients using an LTP induction protocol. Pyramidal neurons in the CA1 region are characterized by relatively low endogenous buffering capacity in the dendrites (~100) and even lower buffering capacity (~20) in the spines [START_REF] Sabatini | The life cycle of Ca 2+ ions in dendritic spines[END_REF]. Thus, the buffering capacity of 300 µM Bis-Fura-2 (K d ~500 in the presence of Mg 2+ , see Table 4.1) is ~6 times larger than the buffering capacity of the dendrite and ~30 times larger than the buffering capacity of the spine. Ca 2+ signals associated either with back-propagating action potentials or with EPSPs were mediated by Ca 2+ influx through voltage-gated Ca 2+ channels and/or through NMDA receptors. As shown in Fig. 4.4a, the pairing of the two stimulating protocols elicited a supra-linear Ca 2+ signal. In the presence of the NMDA receptor blocker AP-5, since the measurements were done using a nonsaturating concentration of a high-affinity indicator, the supra-linear Ca 2+ signal must have been caused by supra-linear Ca 2+ influx mediated by recruitment of additional VGCCs. In agreement with this expectation, at each site where a supra-linear Ca 2+ signal was observed, the V m related optical signal during the pairing protocol had a larger peak depolarization compared to the signals associated with unpaired stimulations (Fig. 4.4a).

In measurements from cerebellar Purkinje neurons (Canepari and Vogt 2008), cells were loaded with 1 mM of the Ca 2+ indicator Fura-FF and with the voltage sensitive dye JPW-1114. Purkinje neurons have an exceptionally high dendritic K cell estimated at ~2000 [START_REF] Fierro | High endogenous calcium buffering in Purkinje cells from rat cerebellar slices[END_REF]. Therefore, the addition of a low affinity Ca 2+ indicator such as Fura-FF (K d ~10), even at millimolar concentrations, does not significantly alter the physiological homeostasis and it is possible to record changes in free intracellular Ca 2+ concentration (∆[Ca 2+ ] i signals (Canepari et al. 2008). In this particular preparation, it was possible to calibrate V m signals in terms of membrane potential. As shown in Fig. 4.4b, pairing the stimulation of the large climbing fibre synaptic potential with local parallel fibres stimulation generated a local dendritic supra-linear ∆[Ca 2+ ] i signal which was independent of Ca 2+ release from stores [START_REF] Brenowitz | Associative short-term synaptic plasticity mediated by endocannabinoids[END_REF]. Although the ∆[Ca 2+ ] i signal originated from Ca 2+ influx through VGCCs, in Purkinje neurons, in contrast to CA1 pyramidal neurons, the V m signal during the pairing protocol did not have larger peak depolarization compared to the V m signals associated with unpaired stimulations (Fig. 4.4b). In this case the Ca 2+ influx associated with the climbing fibre synaptic potential was the same in paired and unpaired conditions. The corresponding ∆[Ca 2+ ] i signal, however, was larger following paired stimulation because the Ca 2+ influx associated with the parallel fibre stimulation locally and transiently saturated the endogenous Ca 2+ buffer (Canepari and Vogt 2008). This supra-linear Ca 2+ signal could be correctly interpreted only when recorded using a low-affinity indicator. When the cell was injected with 10 mM Bis-Fura-2, K dye became ~10 times larger than K cell and supra-linear Ca 2+ signals were abolished (Canepari and Vogt 2008).

Simultaneous V m and Ca 2+ imaging

Sequential V m and Ca 2+ imaging produce valid results if identical responses are evoked in repetitive measurements. Many physiological processes, however, are stochastic, showing large trial-to-trial variability. In these cases, the study of the time variability contains important biological information and simultaneous V m and Ca 2+ optical measurement is required. Simultaneous V m and Ca 2+ imaging was initially achieved using the Ca 2+ indicator Calcium Orange, which has excitation peak at ~550 nm and emission peak at ~580 nm [START_REF] Sinha | Simultaneous optical recording of evoked and spontaneous transients of membrane potential and intracellular calcium concentration with high spatio-temporal resolution[END_REF][START_REF] Sinha | Simultaneous optical recording of membrane potential and intracellular calcium from brain slices[END_REF], or with blue-excitable indicators [START_REF] Bullen | High-speed, random-access fluorescence microscopy: I. Highresolution optical recording with voltage-sensitive dyes and ion indicators[END_REF][START_REF] Bullen | Indicators and optical configuration for simultaneous high-resolution recording of membrane potential and intracellular calcium using laser scanning microscopy[END_REF]. These possibilities may have, however, some inherent problems. First, the small overlap between the emission spectra of the two dyes may produce a "bleeding" of the V m signal into the Ca 2+ recording channel that contaminates the signal. In some situations, such emission cross talk can be mathematically corrected [START_REF] Sinha | Simultaneous optical recording of evoked and spontaneous transients of membrane potential and intracellular calcium concentration with high spatio-temporal resolution[END_REF].

Second, while V m recordings of synaptic or action potentials can be often achieved within short time, the complete time-course of the corresponding Ca 2+ signal may require longer acquisition intervals [START_REF] Canepari | Combining voltage and calcium imaging from neuronal dendrites[END_REF]. In this case, the voltage sensitive dye is exposed to high intensity light for a longer time than what would be required for the V m measurement alone, increasing both bleaching of the dye and phototoxicity. Simultaneous V m and Ca 2+ imaging was also performed using the combination of an absorption voltage sensitive dye with a fluorescent Ca 2+ indicator. Using bath application of the absorption dye RH-482 and a low-affinity Ca 2+ indicator Magnesium Green, Sabatini and Regehr could correlate the Ca 2+ signal underlying synaptic release with the excitation of presynaptic terminals in cerebellar synapses [START_REF] Sabatini | Timing of neurotransmission at fast synapses in the mammalian brain[END_REF][START_REF] Sabatini | Control of Neurotransmitter Release by Presynaptic Waveform at the Granule Cell to Purkinje Cell Synapse[END_REF]. In these experiments, simultaneous Ca 2+ and V m measurements were done by using excitation light wavelength of ~500 nm and recording simultaneously fluorescence from the Ca 2+ indicator Magnesium Green and the transmitted light from voltage-sensitive dye RH-482 at 710 nm from a single location with two photodiodes. The important limitation of this approach is that the absorption measurements from multiple sites on individual neurons are substantially less sensitive compared to fluorescence measurements [START_REF] Antic | Optical signals from neurons with internally applied voltage-sensitive dyes[END_REF]). In the above example, extensive temporal and spatial signal averaging over a large region of the cerebellar slice, which could be obtained with absorption signals, permitted signal detection from presynaptic terminals [START_REF] Sabatini | Control of Neurotransmitter Release by Presynaptic Waveform at the Granule Cell to Purkinje Cell Synapse[END_REF].

The above mentioned problems of simultaneous V m and Ca 2+ imaging using the same excitation wavelength can be overcome by using a Fura indicator. The realization of this measurement, however, required the design of a dedicated apparatus (Vogt et al. 2011a) to optimize the signal-to-noise ratio (S/N). In principle, di-8-ANEPPS and JPW-1114 could be excited by blue light in the configuration shown in Fig. 4.1c. However, narrow-band laser excitation at 532 nm or at 543 nm is associated with much larger sensitivity [START_REF] Canepari | Imaging Inhibitory Synaptic Potentials Using Voltage Sensitive Dyes[END_REF]) permitting single-trial recordings. These excitation wavelengths are longer than the emission peak of Fura indicators. To allow transmission of V m and Ca 2+ emission wavelengths, while reflecting intermediate excitation wavelength (543 nm), the apparatus included a dual-band pass dichroic mirror as shown in the schematic in Fig. 4.5a. Multiple wavelengths dichroic mirrors are commercially available from several companies (e. g. Semrock). The mirror that was used in this case reflected 543 nm and transmitted lower green and red wavelengths corresponding to the emissions of Fura and JPW1114 indicators respectively. This optimal configuration allowed resolving dendritic excitation and associated Ca 2+ signals from relatively small regions in single trial measurements. In the example shown in Fig. 4.5b, simultaneous ∆V m and ∆[Ca 2+ ] i signals, calibrated as described in Canepari and Vogt (2008), associated with one climbing fibre-EPSP and a train of five parallel fibre EPSPs at 100 Hz were compared in two small dendritic regions. ∆V m and ∆[Ca 2+ ] i signals associated with large depolarization could be observed in both regions for the climbing fibre EPSP but only near the parallel fibre stimulating electrode, for the parallel fibre EPSPs train. These measurements, where signals of ~1% change of fluorescence could be resolved from regions of few microns, represent the current state-of-the-art in sensitivity of simultaneous V m and Ca 2+ imaging from single cells.

Simultaneous optical measurement of V m and Ca 2+ currents

Simultaneous V m and Ca 2+ imaging can be also used for the study of native fast Ca 2+ currents based on derivation of Ca-currents from optical signals., We have already stated that the ∆F/F 0 Ca 2+ signal is proportional to the change in concentration of Ca 2+ ions bound to the indicator. If, in addition, the Ca 2+ bound to the indicator is also proportional to the total free Ca 2+ flux entering the cell through the plasma membrane, then the Ca 2+ current can be estimated by the derivative of ∆F/F 0 (Sabatini and Regehr 2008). This estimate is possible with low-affinity indicators [START_REF] Kao | Ca2+ binding kinetics of fura-2 and azo-1 from temperature-jump relaxation measurements[END_REF] where the equilibration of the dye-Ca 2+ binding reaction is faster than the Ca 2+ influx. It was recently demonstrated that the reaction between Ca 2+ and low-affinity indicator Oregon Green 488 BAPTA-5N is faster than a typical VGCC-mediated current, i.e. the ∆F/F 0 signal from this indicator matches the kinetics of the Ca 2+ current integral [START_REF] Jaafari | Imaging Fast Calcium Currents beyond the Limitations of Electrode Techniques[END_REF]. It follows that the kinetics of the ∆F/F 0 derivative is proportional to the kinetics of the Ca 2+ current and can be calibrated in terms of "Ca 2+ current volume density". The simultaneous V m optical measurement allows studying the voltage dependence of the current, i.e. the membrane potential of activation and inactivation of the channel. Technically, simultaneous V m and Ca 2+ imaging using Oregon Green 488 BAPTA-5N requires blue illumination for both indicators (see previous paragraph) and high-speed acquisitions (20 kHz frame rate) to investigate the time-course of the Ca 2+ current at high temporal resolution.

Combining V m imaging with photostimulation

Combining V m imaging with glutamate or GABA photorelease

Membrane potential imaging can be combined with uncaging techniques to investigate how the photorelease of biologically active molecules changes the membrane potential in different regions of a cell. In particular, the caged compounds 4-Methoxy-7-nitroindolinyl-caged-L-glutamate (MNI-glutamate, [START_REF] Canepari | Photochemical and pharmacological evaluation of 7-nitroindolinyl-and 4-methoxy-7-nitroindolinyl-amino acids as novel, fast caged neurotransmitters[END_REF]) and 1-(4-Aminobutanoyl)-4-[1,3-bis(dihydroxyphosphoryloxy)propan-2-yloxy]-7nitroindoline (DPNI-GABA, [START_REF] Trigo | Laser photolysis of DPNI-GABA, a tool for investigating the properties and distribution of GABA receptors and for silencing neurons in situ[END_REF]) can be used to obtain fast photorelease of either Lglutamate or GABA in sub-cellular regions of variable size and to investigate the regulation of membrane potential. An example of this methodological approach is the study of shunting inhibition [START_REF] Mann | Role of GABAergic inhibition in hippocampal network oscillations[END_REF]. Propagation of action potentials in the axon or back-propagation of action potentials in the dendrites can be strongly regulated by local changes of membrane conductance due to opening of GABA A receptors. Simultaneous V m imaging and GABA photorelease is an ideal approach to investigate shunting inhibition. In the CA1 hippocampal pyramidal neuron shown in GABA receptors and the possibility to perform pharmacological studies that cannot be done using presynaptic stimulation. Thus, combining membrane potential imaging with GABA photorelease shows the spatial restriction of the shunting effect and the resulting rapid loss in action potential backpropagation.

Using glutamate uncaging to calibrate voltage-sensitive dyes signals

The calibration of V m optical signals on an absolute scale (in mV) provides direct measurements of V m signal amplitudes from multiple locations. This information can be used to construct and analyze the dynamic spatial maps of signal amplitudes throughout neuronal processes. The temporally and spatially well resolved maps of activity are one of the key elements in understanding the rules governing dendritic signal processing and plasticity. The absolute calibration of optical V m signals, however, is not straightforward and often the exact callibration is not possible. Voltage-sensitive dye recording of membrane potential transients belongs to a class of indirect measurements. The quantity that is being measured directly, by photodetectors, is light intensity, and the quantity that needs to be monitored is derived from a known relationship between the light intensity and V m . It is convenient that the relationship between light intensity and V m is strictly linear over the entire physiological range for many voltage sensitive dyes (e. g. [START_REF] Gupta | Improvements in optical methods for measuring rapid changes in membrane potential[END_REF][START_REF] Loew | Charge-shift probes of membrane potential: a probable electrochromic mechanism for p-aminostyrylpyridinium probes on a hemispherical lipid bilayer[END_REF][START_REF] Wu | Fast multisite optical measurement of membrane potential[END_REF]. In a multisite recording, the fractional change in light intensity is proportional to V m , but also, to a different extent at different sites, to an additional factor: the ratio of inactive dye to active dye. The inactive dye is bound to connective tissue or any other membranes that do not change potential while the active dye is bound to the excitable membrane being monitored. The inactive dye contributes to the resting fluorescence only, and the light from active dye contributes to the resting fluorescence and also carries the signal. In experiments utilizing intracellular application of the dye, inactive dye would be bound to intracellular membranes and organelles. It is a general rule that the ratio of active dye to inactive dye is different for different regions of the object and is also unknown. It follows that the sensitivity of recording from different regions of the object will be different and the calibration of all detectors cannot be done by calibrating the optical signal from any single site. In this situation, the calibration is absolute and straightforward if a calibrating electrical signal that has known amplitude at all locations is available. An all-or-none action potential signal is ideal for this purpose and can be used to create a sensitivity profile of the measuring system. This type of calibration was used to scale the amplitudes of sub-threshold signals in mitral cells of the olfactory bulb [START_REF] Djurisic | Voltage imaging from dendrites of mitral cells: EPSP attenuation and spike trigger zones[END_REF]). If action potentials are not available, another type of calibrating electrical signal can be a slow electrical change spreading with minimal attenuation over relatively long distances. Such condition occurs, for instance, in cerebellar Purkinje neurons (Roth and Hausser 2001;Canepari et al. 2008).

The possibility to achieve uniform depolarization over the entire dendritic tree using L-glutamate photolysis may extend the possibility to calibrate fluorescence signals to other cell types. This calibration procedure is based on the principle that if the ionotropic glutamate receptor becomes the dominant conductance in a particular neuronal compartment, its reversal potential will determine the membrane potential of the compartment. Thus, if dominance of glutamate receptor conductance is obtained by L-glutamate uncaging, the resulting V m change in different compartments of the neuron will be the same and can be used to calibrate voltage-sensitive dye imaging data. The idea of a calibration protocol based on L-glutamate uncaging relies on saturating depolarization in the visualized area (Vogt et al. 2011b). The fractional change of fluorescence will then correspond to the resting membrane potential in each visualized site where photolysis occurs.

In the CA1 hippocampal pyramidal neuron of Fig. 4.8a, one action potential was evoked by somatic current injection. The fractional changes of fluorescence associated with the back-propagating AP from four ROIs on the apical dendrites are shown in Fig. 4.8b (left traces). The amplitude of the signal is variable and could not be directly correlated with the absolute change of membrane potential. After addition of 1 µM TTX, to block action potentials, and of 50 µM CTZ, to prolong depolarization by blocking AMPA receptor desensitization, L-glutamate was photoreleased over the whole field of view.

Combining V m imaging with ChR stimulation

For control of neuronal activity, optogenetic approaches based on light-sensitive channels and other photo-reactive molecules enable highly selective control of genetically-defined populations of neurons [START_REF] Mancuso | Optogenetic probing of functional brain circuitry[END_REF][START_REF] Yizhar | Optogenetics in neural systems[END_REF]. Such techniques offer high time resolution and high spatial precision, allowing stimulation of either individual neurons or large neuronal populations. A commonly used protein for this application is Channelrhodopsin2 (ChR2, [START_REF] Zhang | Channelrhodopsin-2 and optical control of excitable cells[END_REF]. Combination of optical stimulation using ChR2, with membrane potential imaging is in principle a potent approach to investigate electrical activity both at cellular and at network level. The accomplishment of this type of experiment, however, requires voltage-sensitive dyes that are excitable by red wavelengths since activation of ChR2 occurs at visible wavelengths <600 nm. Indicators with these characteristics have been developed by Rina Hildensheim and Amiram Grinvald to investigate electrical activity from the entire cortex in vivo without optical interference of hemoglobin absorption (Grinvald and Hildensheim 2004). More recently, red or near-IR excitable or fluorescent indicators have been developed by Joseph Wuskell, Ping Yan and Leslie Loew to monitor electrical activity in the heart [START_REF] Matiukas | New near-infrared optical probes of cardiac electrical activity[END_REF] and in the brain [START_REF] Wuskell | Synthesis, spectra, delivery and potentiometric responses of new styryl dyes with extended spectral ranges[END_REF] or to be injected in individual cells [START_REF] Zhou | Intracellular long-wavelength voltagesensitive dyes for studying the dynamics of action potentials in axons and thin dendrites[END_REF][START_REF] Yan | Palette of fluorinated voltage-sensitive hemicyanine dyes[END_REF]).

Among these new voltage sensitive dyes, five have been specifically tested to be used in combination with ChR2 stimulation [START_REF] Kee | Imaging activity of neuronal populations with new long-wavelength voltage-sensitive dyes[END_REF]. In the example shown in Fig. 4.9a (modified from [START_REF] Tsuda | Probing the function of neuronal populations: combining micromirror-based optogenetic photostimulation with voltage-sensitive dye imaging[END_REF]), ChR2 was selectively expressed in interneurons located in the molecular layer, providing inhibitory post-synaptic potentials (IPSPs) to cerebellar Purkinje neurons when the light-sensitive channel was excited. In this experiment, the cerebellar slice was stained with the voltage sensitive dye Di-2-ANBDQPQ [START_REF] Zhou | Intracellular long-wavelength voltagesensitive dyes for studying the dynamics of action potentials in axons and thin dendrites[END_REF]) and excited at 658 nm, beyond the excitation spectrum of ChR2 [START_REF] Kee | Imaging activity of neuronal populations with new long-wavelength voltage-sensitive dyes[END_REF]. Transient photo-stimulation of the molecular layer with blue light produced an IPSP in the nearby Purkinje neurons. As shown in Fig. 4.9a, the IPSP elicited by photo-stimulation of the molecular layer could be measured optically. Notably, the kinetics of the optical signal was identical to that of the IPSP measured with a patch electrode. In another study, ChR2 was selectively expressed in GABAergic interneurons of the hippocampal stratum lacunosum moleculare and the regulation of electrical activity in different areas was characterized by staining the slice with the indicator DI-4-ANBDQPQ [START_REF] Kee | Imaging activity of neuronal populations with new long-wavelength voltage-sensitive dyes[END_REF] and by performing V m imaging [START_REF] Leão | OLM interneurons differentially modulate CA3 and entorhinal inputs to hippocampal CA1 neurons[END_REF]). Thus, using this strategy, it was possible to determine that interneurons from the stratum lacunosum molecular differentially modulate CA3 and entorhinal inputs to hippocampal CA1 neurons.

Using water soluble indicators excitable with red light [START_REF] Zhou | Intracellular long-wavelength voltagesensitive dyes for studying the dynamics of action potentials in axons and thin dendrites[END_REF][START_REF] Yan | Palette of fluorinated voltage-sensitive hemicyanine dyes[END_REF], electrical activity can be imaged in combination with ChR2 photostimulation from individually stained cells [START_REF] Willadt | Combined Optogenetics and Voltage Sensitive Dye Imaging at Single Cell Resolution[END_REF]. The clear advantage of this approach is that the higher sensitivity of V m imaging in individually stained neurons permits optical measurements of smaller membrane potential changes from well identified sub-cellular compartments. In the examples shown in Fig. 4.9b, ChR2 was expressed by all GABAergic interneurons and individual hippocampal cells were filled with the indicator Di-2-ANBDQPTEA. When electrical and optical recordings were performed from a GABAergic interneuron, ChR2 photo-stimulation of the recorded cell generated an action potential that could be monitored both electrically and optically from proximal dendritic segment (Fig. 4.9b,left). In contrast, when electrical and optical recordings were performed from a CA1 pyramidal neuron, ChR2 photo-stimulation of presynaptic interneurons generated an IPSP that was recorded both electrically and optically from the proximal part of the apical dendrite (Fig. 4.9b, right). In conclusion, these initial studies indicate that combining membrane potential imaging with optogenetic photostimulation is likely to become a standard approach in the near future.

Fig. 4 .

 4 Fig. 4.1 shows three examples of spectral representations for the different cases and the optical arrangements to combine the recording of the two signals. In the first case (Fig. 4.1a), the fluorescencefrom the two dyes can be excited using two separate wavelengths and the emitted light from the two indicators is recorded using a single detector. In this case, it is never possible to achieve a simultaneous measurement since the imaging experiment requires alternating between the two different excitation wavelengths. In the second case (Fig.4.1b), because the excitation spectra are overlapping, fluorescence from both indicators can be excited using the same excitation wavelength. Emitted light can be recorded by two different detectors after splitting the emitted light with a secondary dichroic mirror. Thus, by synchronizing data acquisition of the two detection systems, it is possible to achieve simultaneous imaging of the fluorescence signals from the two indicators. Finally, in the third case (Fig.4.1c), the combined excitation and detection of the two fluorescence systems can be done using two excitation wavelengths and two detection systems simultaneously.

  signals. An example of V m and Ca 2+ optical signals recorded sequentially from the two indicators in three different regions on the dendritic tree of a Purkinje neuron stained by JPW-1114 and Fura-FF is illustrated in Fig. 4.3. Using an excitation band of 387±5 nm for Fura-FF (Fig. 4.3a), the Ca 2+ increase associated with a climbing fibre excitatory postsynaptic potential (EPSP) corresponds to a negative ∆F/F 0

  comparing individual recordings as shown in Fig. 4.3c. In this example, the V m and the Ca 2+ optical signals were recorded from a dendritic location on a cerebellar Purkinje neuron in response to four repetitions of climbing fibre activation separated by 1 minute. The results showed that signals were practically identical in 4 individual trials (gray traces) permitting the correlation of V m and Ca 2+ optical signals recorded sequentially in response to the same stimulus.

  Fig. 4.6a shows an example of a CA1 hippocampal pyramidal neuron filled with 1 mM of Oregon Green 488 BAPTA-5N and the voltage sensitive dye JPW1114. Fig. 4.6b shows the Ca 2+ current volume density (I Ca /V) associated with an action potential monitored optically in the proximal apical dendritic segment of a CA1 hippocampal pyramidal neuron. The I Ca /V and V m signals were measured under two conditions: for an action potential starting at V m = -60 mV and for an action potential starting at V m = -80 mV (Fig. 4.6b). When the action potential started at V m = -80 mV, I Ca /V was larger and started at a more hyperpolarized V m (Fig. 4.6c), indicating that at V m = -80 mV Ca 2+ current contained a component mediated by low-voltage activated (T-type) Ca 2+ channels (Cueni et al. 2009). The example shown in Fig. 4.6 illustrates how simultaneous optical measurement of V m and Ca 2+ currents can drastically improve our understanding of the physiological function of Ca 2+ channels by permitting the exploration of local biophysics of native channels during physiological activity.

  Fig 4.7a, the effect of local GABA photorelease on the back-propagating action potential in the apical dendrite was explored. DPNI-GABA was applied using a patch pipette positioned close to the distal part of the dendrite at ~200 µm from the soma. The fluorescence change associated with a back-propagating action potential under control conditions (Fig. 4.7b, left traces) was compared with a back-propagating action potential occurring 15 ms after an episode of GABA photorelease (Fig. 4.7b, right traces) from four regions of interest. The amplitude of the somatic action potential was the same in both cases, but local GABA uncaging decreased the amplitude of the fluorescence change associated with the backpropagating action potential progressively with the distance from the soma. Although shunting inhibition induced by GABA uncaging is produced by non-physiological co-activation of synaptic and extrasynaptic GABA receptors, this experiment represents a useful model of this phenomenon since it permits a clear separation of presynaptic and postsynaptic effects, a clear localization of the activated

  The recording was performed starting from the resting V m of the cell. The size of the fluorescence change at each site and the absolute somatic depolarization elicited by uncaging increased with the amount of released L-glutamate, i.e. with the exposure time of the UV light, eventually reaching saturation. The saturating V m corresponded to the reversal potential of glutamate receptors. Using this information, the amplitude of the back-propagating action potential signals was converted into the change in membrane potential. This method of calibration of voltage-sensitive dyes signals, based on Lglutamate photorelease, can only be applied to cell types characterized by dominant glutamate conductance.

Figure 4 . 1 .

 41 Figure 4.1. Alternatives for combined imaging using two-indicators. a) (Left) Ideal absorption and emission spectra of two indicators with separate absorption spectra and overlapping emission spectra. Arrows indicated by "ex" and "em" are positioned over ideal excitation and emission. Rectangles indicated by "d" are in position of ideal dichroic mirrors. (Right) Optical arrangements for excitation and recording of fluorescence using either one or two light sources. b) (Left) Same as a) but for two indicators with overlapping absorption spectra and separate emission spectra. (Right) Optical arrangements for excitation and recording of fluorescence using either one or two light detection systems. c) (Left) Same as a) and b) but for two indicators with separate absorption and emission spectra. (Right) Optical arrangements for excitation and recording of fluorescence using either one or two light sources and light detection systems.

Figure 4 . 2 .

 42 Figure 4.2. Absorption and emission spectra of di-8 ANEPPS in combination with the ideal absorption and emission of Calcium Green and Fura calcium indicators. Left and right curves are the absorption and emission spectra as reported by Invitrogen -Molecular Probes. The thick downward arrow represents the excitation of the ratiometric indicator Fura. The thin downward arrows represent the excitation peak of Calcium Green. The upward arrows represent the emission peaks of the two indicators.

Figure 4 . 3 .

 43 Figure 4.3. Combining V m and Ca 2+ imaging using JPW-1114 and a UV excitable Ca 2+ indicator. a) Schematic of the imaging apparatus for sequential V m and Ca 2+ imaging using a Fura indicator. Filters for voltage-sensitive dye (JPW): ex1 = 525±25 nm, d1 > 570 nm and em1 > 610 nm. Filters for Ca 2+ indicator (Fura): ex2 = 387±6 nm, d1 > 470 nm and em1 = 510±42 nm. b) V m and Ca 2+ fractional changes of fluorescence from cerebellar Purkinje neuron dendrites related to a climbing fibre EPSP recorded from the locations 1-3 reported in the fluorescence images on the left. Somatic recording of the climbing fibre EPSP is the upper trace. c) Individual V m (left) and Ca 2+ (right) fractional changes of fluorescence (gray traces) related to a climbing fibre EPSP. Superimposed black traces are the averages of the four trials.

Figure 4 . 4 .

 44 Figure 4.4. Combining V m and Ca 2+ imaging using high-and low-affinity Fura indicators. a) (Left) Ca 2+ and V m fractional changes of fluorescence corresponding to back-propagating action potentials (bAPs) and paired EPSP-bAP signals from two locations on the dendritic tree of a CA1 hippocampal pyramidal

Figure 4 . 5 .

 45 Figure 4.5. Simultaneous V m and Ca 2+ imaging. a) Schematic of the optical configuration to achieve simultaneous V m and Ca 2+ imaging; 543 nm and 387 nm are used for voltage sensitive dye and Fura indicator excitations respectively; low green and red emissions of the two indicators are separated and detected with two cameras. b) (Top) Reconstruction of a cerebellar Purkinje neuron with the dendritic area in recording position outlined; two sites of interest (1 and 2) are indicated by the arrows. (Bottom) ∆V m (red traces) and ∆[Ca 2+ ] i (blue traces) signals associated with a climbing EPSP (left) or with a train of five parallel fibre EPSPs at 100 Hz (right) from the two sites of interest; somatic electrical recordings shown below. Modified from Vogt et al. 2011a.

Figure 4 . 6 .

 46 Figure 4.6. Dendritic Ca 2+ currents associated with an action potential at different starting V m . a) Fluorescence images of a CA1 hippocampal pyramidal neuron filled with Oregon Green 488 BAPTA-5N (left image) and the voltage sensitive dye JPW1114 (right image); the region of interest from where fluorescence was averaged is outlined. b) Ca 2+ currents (blue) associated with one action potential starting either at V m = -60 mV or -80 mV; V m recorded optically (red) and with the electrode (black) superimposed. c) Ca 2+ currents normalized and superimposed to the action potential in the two cases; V m at the current onset (V m 0 ) indicated. Modified from Jaafari et al. 2014.

Figure 4 . 7 .

 47 Figure 4.7. Investigating shunting inhibition using combined V m imaging and local GABA photorelease. a) Image of a CA1 hippocampal pyramidal neuron; the dendritic area in recording position is outlined and four regions of interest are shown on the right; the position of the pipette used for caged-glutamate application is illustrated. b) V m optical signals from the regions 1-4 and somatic recordings associated

Figure 4 . 8 .

 48 Figure 4.8. Calibration of fractional changes of fluorescence in terms of membrane potential using Lglutamate a) Image of a CA1 hippocampal pyramidal neuron; the dendritic area in recording position is outlined and four regions of interest are shown on the right; caged-glutamate is applied over the visualized area of the dendrite and wide-field illumination was used for photolysis. b) (Left) V m optical signals from the regions of interest 1-4 and somatic recordings associated with one back-propagating action potential. (Right) After addition of 1 µM TTX and 50 µM CTZ, V m optical signals from the same regions and somatic recordings associated with 1 ms L-glutamate photo-release. The peaks of the somatic signal are indicated over their respective traces; the sizes of the estimated peak spikes in the regions of interest are indicated over their respective traces. Modified from Vogt et al. 2011b.

Figure 4 . 9 .

 49 Figure 4.9. Combining V m imaging with ChR2 stimulation. a) (Left) Schematic of the optical configuration to achieve this combination; red and blue illumination are used for voltage sensitive dye and ChR2 excitations respectively; IR fluorescence emission is detected. (Right) IPSP recorded electrically (with a patch electrode) and optically from a the soma of a Purkinje neuron following ChR2 excitation at interneurons located in the molecular layer; modified from Tsuda et al. 2013 b) Imaging of light evoked action potential in a ChR2 expressing interneuron (left) and of IPSP in CA1 pyramidal neuron induced by ChR stimulation at interneurons (right). (Top) Schematic illustrations of the experiment on the top with site of blue flash and imaging region. (Bottom) Simultaneous optical dendritic and electrical somatic recordings; arrows indicate blue light application (5 ms); modified from Willadt et al. 2014.

Table 4

 4

	.1: Dissociation constants and buffering capacities of Fura dyes. A list of commercially available Fura dyes with dissociation constant (K d ) and buffering capacities (K dye ) at the two concentrations of 300 µM and 1 mM. References: (a)