Synchrony of Fast-Spiking Interneurons ... - Semantic Scholar

LETTER

Communicated by Carson Chow

Synchrony of Fast-Spiking Interneurons Interconnected by GABAergic and Electrical Synapses Masaki Nomura [email protected] Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan, and CREST, Japan Science and Technology Corporation, Kawaguchi, Saitama 332-0012, Japan

Tomoki Fukai [email protected] Department of Information-Communication Engineering, Tamagawa University, Tokyo 194-8610, Japan; CREST, Japan Science and Technology Corporation, Kawaguchi, Saitama 332-0012, Japan

Toshio Aoyagi [email protected] Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan, and CREST, Japan Science and Technology Corporation, Kawaguchi, Saitama 332-0012, Japan

Fast-spiking (FS) interneurons have specific types (Kv3.1/3.2 type) of the delayed potassium channel, which differ from the conventional HodgkinHuxley (HH) type potassium channel (Kv1.3 type) in several aspects. In this study, we show dramatic effects of the Kv3.1/3.2 potassium channel on the synchronization of the FS interneurons. We show analytically that two identical electrically coupled FS interneurons modeled with Kv3.1/3.2 channel fire synchronously at arbitrary firing frequencies, unlike similarly coupled FS neurons modeled with Kv1.3 channel that show frequency-dependent synchronous and antisynchronous firing states. Introducing GABAA receptor-mediated synaptic connections into an FS neuron pair tends to induce an antisynchronous firing state, even if the chemical synapses are bidirectional. Accordingly, an FS neuron pair connected simultaneously by electrical and chemical synapses achieves both synchronous firing state and antisynchronous firing state in a physiologically plausible range of the conductance ratio between electrical and chemical synapses. Moreover, we find that a large-scale network of FS interneurons connected by gap junctions and bidirectional GABAergic synapses shows similar bistability in the range of gamma frequencies (30–70 Hz).

c 2003 Massachusetts Institute of Technology Neural Computation 15, 2179–2198 (2003)

2180

M. Nomura, T. Fukai, and T. Aoyagi

1 Introduction Results of many recent experiments suggest that synchronous neuronal oscillations play important functional roles in the brain (Buzsáki, Llinás, Singer, Berthoz, & Christen, 1994; Singer & Gray, 1995; Gerstner, van Hemmen, & Cowan, 1996). Although the neural mechanism of such synchronous oscillations has not been fully clarified, it appears likely that networks of inhibitory interneurons promote synchrony in local cortical circuitry (Lytton & Sejnowski, 1991; Cobb, Huhl, Halasy, Paulsen, & Somogyi, 1995; Whittington, Traub, & Jefferys, 1995; Beierlein, Gibson, & Connors, 2000). Fastspiking (FS) cells and low-threshold-spiking (LTS) cells are considered to be two major categories of GABAergic interneurons in the neocortex (Gibson, Beierlein, & Connors, 1999). It has been found that nearby pairs of interneurons belonging to the same class are often interconnected simultaneously by electrical synapses (gap junctions) and chemical (GABAergic) synapses, the latter of which are sometimes bidirectional (Galarreta & Hestrin, 1999, 2001, 2002; Gibson et al., 1999). Modeling a network of FS interneurons, we explore the combined effects of gap junctions and GABAergic synapses on the synchronous firing of the interneurons. A characteristic feature of FS interneurons is the appearance of sustained high-frequency trains of brief action potentials with little spike frequency adaptation (Connors & Gutnick, 1990). As was recently revealed (Erisir, Lau, Rudy, & Leonard, 1999; Rudy & McBain, 2001), it is the Kv3.1/3.2 voltagegated K+ channels, but not the Kv1.3 voltage-gated K+ channels, that play a crucial role in creating this feature. Positively shifted voltage dependence and fast deactivation rates of the Kv3.1/3.2 potassium current enable the FS neurons to discharge at high frequencies. Several theoretical studies have been conducted on networks of inhibitory interneurons (White, Chow, Ritt, Soto-Trevino, ˜ & Kopell, 1998; Chow, White, Ritt, & Kopell, 1998; Traub et al., 2001; Suzuki & Aoyagi, 2002), but in these studies, the characteristic electronic properties of FS cells have not been fully taken into account. Many previous studies modeled both interneurons and pyramidal cells based on the Hodgkin-Huxley (HH) neurons that incorporate the Kv1.3 K+ channel. It was shown, however, that synchronous spiking behavior of simple model neurons significantly depends on the neuronal membrane properties (Hansel, Mato, & Meunier, 1995; Ermentrout, 1996). For instance, Sherman and Rinzel (1992) found that a strong gap junction can synchronize two electrically coupled neurons, but a weak gap junction may induce various complicated synchronization patterns depending on the membrane properties. Chow and Kopell (2000) elucidated that spike shape and size play a crucial role in stabilizing synchronous firing states of two electrically coupled neurons. In this article, we show that the incorporation of Kv3.1/3.2 K+ channels dramatically changes the dynamic behavior of coupled FS interneurons. To elucidate the different roles played by gap junctions and GABAergic synapses in the emergence of synchronous firing, we first study how

Synchrony of Fast-Spiking Neurons

2181

spikes of two weakly coupled FS interneurons are synchronized or desynchronized by each of the two synaptic transmissions. These studies are conducted analytically by means of the phase-reduction method (Kuramoto, 1984; Ermentrout & Kopell, 1984; van Vreeswijk, Abbott, & Ermentrout, 1994; Hansel et al., 1995). Although this method was formulated in the limit of weak coupling, it does provide some insight into the synchronizing behavior of not so weakly coupled neuron pairs and that of a large-scale neural network. Since the GABAergic synapses can be either bidirectional or unidirectional between an FS interneuron pair (Galarreta & Hestrin, 1999, 2001, 2002; Gibson et al., 1999), we study the synchronizing behavior with both unidirectional and bidirectional GABAergic synaptic connections. Somewhat unexpectedly, near-synchronous firing is a unique stable state with a gap junction and a unidirectional GABAergic synapse. In contrast, combinations of a gap junction and a bidirectional GABAergic synapse allow both a synchronous and antisynchronous firing state at physiologically acceptable values of the decay time constant of the GABAergic synaptic current (< 5 ms). Consistent with these results, a large-scale network of the FS interneurons interconnected by gap junctions and GABAergic synapses tends to exhibit bistability between synchronous firing and non-phase-locked firing. Thus, coherence of the activity may provide an important dimension of information representation in a network of FS interneurons. 2 Model 2.1 Fast-Spiking Interneurons. Many models of interneurons have been proposed with the aim of characterizing their electrophysiological properties and elucidating their roles in various cortical functions (Kanemasa, Gan, Perney, Wang, & Kaczmarek, 1995; Erisir et al., 1999; Durstewitz, Seamans, & Sejnowski, 2000). Erisir et al. carried out pharmacological experiments on the interneurons of the mouse somatosentory cortex and found that Kv3.1/3.2 voltage-gated K+ channels play significant roles in creating a characteristic feature of FS cells: sustained high-frequency firing with little spike frequency adaptation. In this study, we adopt a model of FS neurons that is capable of describing this characteristic feature, which was proposed by Erisir et al. (1999). The behavior of a single FS interneuron is described as follows: C

dV = −gNa m3 h(V − VNa ) − gK n2 (V − VK ) dt − gl (V − Vl ) + Iext + Igap + Isyn , x∞ (V) − x dx = , dt τx (V)

x∞ (V) =

αx , αx + βx

(x = m, h, n),

τx (V) =

1 , αx + βx

(2.1) (2.2) (2.3)

2182


40.0(75.0 − V) , exp[(75.0 − V)/13.5] − 1.0 βm = 1.2262 exp(−V/42.248),

αm =

(2.4)

αh = 0.0035 exp(−V/24.186), βh =

0.017(−51.25 − V) , exp[(−51.25 − V)/5.2] − 1.0

αn =

95.0 − V , exp[(95.0 − V)/11.8] − 1.0

βn = 0.025 exp(−V/22.222),

(2.5)

(2.6)

where V is the membrane potential, m and h are the activation and inactivation variables of the sodium channel, respectively, and n is the activation variable of the potassium channel. Here, the parameter values used are VNa = 55.0 mV, VK = −97.0 mV, Vl = −70.0 mV, gNa = 112 mS/cm2 , gK = 224 mS/cm2 , gl = 0.1 mS/cm2 , and C = 1.0 µF/cm2 . In Figure 1a, we display the time course of the potassium current in this model and that in the HH model with a conventional Kv1.3 K+ channel for comparison. As seen in the figure, Kv3.1 K+ current begins to rise only after the membrane potential reaches its peak, whereas Kv1.3 K+ current conducts during the rising phase of the action potential (Rudy & McBain, 2001). In Figure 1b, we display the current versus frequency relationship for the two models. These results are in accordance with the previous studies (Rinzel & Ermentrout, 1989; Ermentrout, 1996). As seen in the figure, the Kv3-based model of an FS interneuron is capable of firing at very high frequencies (>200 Hz) in response to an external current, whereas in the HH model, the firing frequency is saturated at about 170 Hz. The reason for these differences is well known in the bifurcation theories: transitions to firing occur through a saddle-node bifurcation in the Kv3-based model, whereas the HH model undergoes a Hopf bifurcation (Rinzel & Ermentrout, 1989). In addition, the different bifurcation types may lead to different synchronization properties for an FS interneuron network and a HH neuron network (Hansel et al., 1995; Ermentrout, 1996). From Figure 1b, it is found that the present Kv3-based model neuron responds at gamma frequencies (30–70 Hz) to an external current density ranging from 1 to 4 µA/cm2 . Figure 1c displays the phase-response curves ZV calculated by means of the phase-reduction method (see appendix A) for the FS model firing at 64 Hz (Iext = 3.0 µA/cm2 ) and the HH model firing at 68 Hz (Iext = 10.0 µA/cm2 ). 2.2 Synaptic Couplings. In our FS-neuron network, any FS neuron pair is interconnected by a gap junction as well as by bidirectional or a unidirec-


Kv3.1 current membrane V

20

500

-60

0 1000

HH neuron Kv1.3 current membrane V

-20

300

1000

-20

60 20

(b)

1500

FS neuron

500

frequency [Hz]

60

IK [µA/cm2]

membrane V [mV]

(a)

2183

250

150 100

-60 0

10

0 30

20

50 0

time [ms] (c)

FS neuron HH neuron

200

0

20

γ -range

40

Iext [µA/cm2]

0.6

FS neuron HH neuron

0.4

ZV

0.2 0

-0.2 π

0

2π

φ

(d) membrane V [mV]

50 0 -50

Gap Junction -55 -60 -65

15.3

Iext

-69.0 -69.5 -70.0 -70.5

Synaptic Coupling 1.85

0

10

20

30

40

50

60

time [ms]

70

80

90

100

Iext

Figure 1: (a) Time course of the potassium current of our FS model (top) and that of the HH model (bottom). Vertical broken lines indicate the time when the potassium currents begin to rise. (b) Firing frequencies of our FS model and those of the HH model as functions of the injected current Iext . (c) Examples of the phase-response function ZV for the FS model and the HH model. The response curves are plotted over a period of oscillation. (d) A presynaptic neuron firing at 44 Hz (Iext = 1.7 µA/cm2 ) (top) produces postsynaptic potentials through the gap junction (middle) and the GABAergic synapse (bottom). With the same values of the maximal conductances, the postsynaptic potential through the gap junction is about eight times larger than that through the GABAergic synapse.

2184


tional GABAergic synapse. For the gap junction, we use the equation Igap = −ggap (V − Vpre ),

(2.7)

where Vpre represents the membrane potential of the presynaptic neuron. The GABAergic synapses are described by a kinetic model of receptor binding (Destexhe, Mainen, & Sejnowski, 1994) as Isyn = −gsyn r(V − Vsyn ),

(2.8)

r∞ (T) − r dr = , dt τr (T) r∞ (T) = T=

αr T , αr T + βr

(2.9) τr (T) =

1 , 1.0 + exp(−Vpre )

1.0 , αr T + βr αr =

1.0 − βr , τraise

(2.10) βr =

1.0 τdecay

,

(2.11)

where τraise = 0.1 ms and Vsyn = −75.0 mV for the inhibitory chemical connections mediated by GABAA receptors. Throughout this article, we set the decay time constant as τdecay = 5 ms. It has been shown that the subtypes of the GABAA receptors expressed at the interneuron-to-interneuron synapses are different from those expressed at the interneuron-to-pyramidal cell synapses. The decay time constant of the inhibitory synapses between interneurons can be as small as 2 ms (Bartos et al., 2002; Galarreta & Hestrin, 2002). As seen later, however, any decay time constant less than 6–7 ms gives qualitatively the same results. In the above equations, r can be interpreted as the fraction of the receptors bound to synaptic transmitters. Thus, the value r = 1 (r = 0) means that all channels are open (closed). In Figure 1d, we plot the postsynaptic potentials (PSPs) created by the synaptic transmissions through our models of the gap junction and the GABAergic synapse. With equal magnitudes of the maximal conductances (gsyn = ggap = 0.1 mS/cm2 ), the PSP amplitude induced by the gap junction is about eight times larger than the PSP amplitude induced by the GABAergic synapse. 3 Two Weakly Coupled Neurons 3.1 Electrical Synapse. To begin, we consider the synchronous spiking behavior of the two-neuron networks interconnected by a gap junction (see Figure 2a). When two FS neurons discharge at 64 Hz, their spikes are synchronized at sufficiently large times, as displayed in Figure 2b.1. In contrast, two HH neurons firing at a similar frequency (68 Hz) can evolve into two distinct states depending on the initial conditions (i.e., the neural network shows bistability). If the initial phase difference is smaller than a certain critical value, the HH-neuron pair develops synchrony in a steady state (see


(a)

Gap Junction

Iext

Iext (b1) FS neuron

(b2)

4

50

− ΓSYM

membrane V [mV]

2185

0 -50

30

60

90 120 150 180 210 240 270 300 330

time [ms]

stable unstable

-4

π

0

2π

φ

(c2)

4

− ΓSYM

40 20 0 -20 -40 -60

HH neuron

2 0

stable unstable

-2 -4 0

30

60

90 120 150 180 210 240 270 300 330

time [ms]

(d)

π

0

2π

φ

π FS neuron HH neuron

φ

membrane V [mV]

membrane V [mV]

(c1) HH neuron

40 20 0 -20 -40 -60 -80

0

-2

-100 0

-80

FS neuron

2

0

30

60

90 120 150 180 210 240 270 300 330

time [ms]

0 0

50

100

150

200

250

frequency [Hz]

Figure 2: (a) A two-neuron network coupled through a gap junction. The gap junction is bidirectional. Iext is a common external current. (b.1) Synchronous firing is realized asymptotically in an electrically coupled two-FS neuron network. Here, Iext = 3.0 µA/cm2 (64 Hz) and ggap = 0.01 mS/cm2 . (b.2) Interaction − function SYM for the network used in Figure b.1. The black circles and the gray circle are stable fixed points and an unstable fixed point, respectively. (c.1) Both synchronous firing and antisynchronous firing can be realized asymptotically in an electrically coupled two-HH neuron network. Here, Iext = 10.0 µA/cm2 − (68 Hz) and ggap = 0.01 mS/cm2 . (c.2) Interaction function SYM for the network used in Figure c.1. Black and gray circles have the same meaning as in Figure b.2. (d) Relationship between the stable steady-state phase difference and the firing rate of the FS or HH neurons.

2186


the top row of Figure 2c.1). Contrastingly, when the initial phase difference is larger than this critical value, the final state shows antisynchrony, in which the two HH neurons are phase-locked with a phase difference of π (see the bottom row of Figure 2c.1). Figures 2b.2 and 2c.2 display the inter− action functions SYM of the networks of FS-neuron pair and HH-neuron pair, respectively. Figure 2b.2 indicates that the final state of the FS-neuron network is synchronous, whereas Figure 2c.2 shows that the final state of the HH-neuron network is either synchrony or antisynchrony (bistability). Therefore, these theoretical results are in accordance with the numerical results shown in Figures 2b.1 and 2c.1. In Figure 2d, we display the analytically obtained relationship between the stable steady-state phase difference and the firing rate of the neurons. These results show that a pair of FS neurons is synchronized at arbitrary (from the minimum to the maximum, (∼250 Hz)) firing rates. In contrast, a pair of HH neurons is synchronized only at firing rates larger than a critical value of about 103 Hz. Below this critical frequency (and above spike threshold), the neuron pair exhibits the bistability. It is also known that two electrically coupled leaky integrateand-fire neuron models show synchrony and anti-synchrony depending on firing frequency (Chow & Kopell, 2000). 3.2 Chemical Synapse. Applying the phase reduction method, we investigated the spiking behavior of two FS neurons interconnected by bidirectional GABAergic synapses, as illustrated in Figure 3a. The intensity of the external current was adjusted such that the FS neurons fired at 44 Hz (Iext = 1.7 µA/cm2 ). Figure 3b displays the relationship between the steadystate phase difference and the decay time constant of the GABAergic synaptic current, τdecay . If τdecay < 7.2 ms, only antisynchronous state is stable. Otherwise both synchronous and antisynchronous states are stable. Results of recent experiments suggest that the decay time constant of the GABAA receptor-mediated synaptic currents between FS neurons is smaller than this critical value (Bartos et al., 2002; Galarreta & Hestrin, 2002). Therefore, it is likely that bidirectional GABAergic synapses induce an antisynchronous state in a pair of FS interneurons. We set as τdecay = 5 ms throughout this article. We have confirmed that with bidirectional GABAergic synapses alone, only antisynchronous state is stable for τdecay = 5 ms in the entire range of gamma oscillation. 3.3 Electrical and Chemical Synapses. In this section, we explore the combined effects of gap junctions and GABAergic synapses on the synchronous firing of two FS neurons. Calculating the phase-response curve ZV − and the interaction function (A)SYM , we obtained the steady-state phase difference as a function of the conductance ratio gsyn /ggap , where gsyn and ggap are the maximum conductances of the chemical and electrical synapses, respectively (see appendix A). The results are displayed in Figures 4a.2–4a.3 and 4b.2–4b.3. In Figure 4a.2, we find that only the synchronous state is


2187

Synaptic Coupling

(a)

Iext (b)

Iext

2π stable unstable

φ

π

0

0

5

10

τ 7.19

15

20

decay

Figure 3: (a) A two-neuron network with bidirectional GABAergic synapses. (b) Relationship between the steady-state phase difference and the decay time constant of the GABAergic synaptic current. Stable and unstable phase differences are represented by the black and the gray solid lines, respectively.

stable for weak bidirectional GABAergic synapses (gsyn /ggap < 4.2) and that only the antisynchronous state is stable for strong GABAergic synapses (gsyn /ggap > 24.7). These results are consistent with the findings in the < preceding two sections. Interestingly, for 4.2 < ∼ gsyn /ggap ∼ 24.7, the synchronous state and the antisynchronous state are simultaneously stable. Therefore, in this range of the conductance ratio, the steady state of the network depends on an initial state, as in Figure 2c.1. As the current intensity is increased, the synchronous state is stabilized in an enlarged region and the bistability appears in a broader range of the conductance ratio (see Figure 4a.3). In the case of unidirectional GABAergic synapses, the steady state has a finite and unique phase difference at a given value of gsyn /ggap . A synchronous state can be achieved only when gsyn ∼ 0 mS/cm2 .

2188


(a1)

(a2) I =1.7 [µA/cm2] (44 Hz) (a3) ext

Synaptic Coupling

φ

2π

Gap Junction

stable unstable

π 0 0

Iext

Iext

10

(b2)

(b1) Synaptic Coupling

Iext

20

30

40

gsyn/ggap

π

50

20

30

40

gsyn/ggap

50

Iext=3.0 (64 Hz)

π

φ

0 0

10

2π

π

Iext

0 0

(b3)

Iext=1.7 (44 Hz)

2π

Gap Junction

Iext=3.0 (64 Hz)

2π

10

20

30

40

gsyn/ggap

50

0 0

10

20

30

40

50

gsyn/ggap

Figure 4: (a.1, b.1) Two-neuron networks interconnected simultaneously by a gap junction and bidirectional (a.1) or unidirectional (b.1) GABAergic synapses. (a.2, a.3, b.2, b.3) Phase diagrams obtained by the phase-reduction method as functions of the conductance ratio gsyn /ggap . The top panels are for the symmetric network, and the bottom ones are for the asymmetric network. Two different intensities of the external current were adopted: Iext = 1.7 and 3.0 µA/cm2 , which give firing rates of 44 Hz and 64 Hz, respectively. Stable and unstable phase differences are represented by black and gray solid curves, respectively. The arrows in a.2 and b.2 indicate a physiologically plausible value of the conductance ratio.

4 Large-Scale Networks 4.1 Bidirectional Chemical Synapses. In this section, we study phaselocking phenomena in a network consisting of 100 FS interneurons connected all-to-all by weak GABAergic synapses and gap junctions (a fully coupled network). First, we investigate the case of symmetric GABAergic connections. To examine whether the large-scale network shows an initial condition-dependent behavior that indicates bistability, we set the parameters to those values that yield bistability in a symmetrically connected two-neuron network (e.g., Iext = 1.7 µA/cm2 and gsyn /ggap = 10; see Figure 4a.2). In numerical computations, spikes were synchronized if the initial phases were chosen randomly and independently from a gaussian distribution with a sufficiently small standard deviation (see Figure 5a.1). Contrastingly, the same network settled into a two-cluster state if the initial phases were chosen randomly and independently from a uniform distribution (see Figure 5a.2). This two-cluster state is the counterpart of the antisynchronous state in the two-neuron network. If the coherence of the initial


(a1)

100

80

80

neuron #

100

neuron #

60 40

40

0 100

0 100

50 0

0

50

5050

time [ms]

20100

histogram

20

50 0

0

(b2)

100

100

80

80

neuron #

neuron #

60

20

60 40

0 100

0 100

0

0

50

13050

time [ms]

25100

13050

25100

50

13050

25100

time [ms]

40 20

50

50

60

20

histogram

histogram

(a2)

(b1)

histogram

2189

50 0

0

time [ms]

Figure 5: The initial condition-dependent and the network structure-dependent activities of the large-scale networks of FS interneurons. In each figure, top panels display the raster plots representing the activity of 100 FS interneurons, and bottom panels show the spike histograms at each instant. The bin width is 1 ms. Values of the parameters were set as Iext = 1.7 µA/cm2 and gsyn /ggap = 10, such that the conductance ratio takes a physiologically reasonable value (gsyn = 10−4 mS/cm2 and ggap = 10−5 mS/cm2 ). (a.1, a.2) The neurons are fully and symmetrically connected by the GABAergic synapses. (b.1, b.2) Among all the neuron pairs, 30% are connected bidirectionally, and the rest are connected unidirectionally in randomly determined directions (70% asymmetric network).

2190


state was low, the steady state of the network was in general a two-cluster state. Both the synchronous state and two-cluster state could be reached in almost the entire range of gamma oscillation. A linear stability analysis (Kaneko, 1990; Golomb, Hansel, Shraiman, & Sompolinsky, 1992; Hansel, Mato, & Meunier, 1993; Okuda, 1993; Kori & Kuramoto, 2001) has confirmed the stability of the two-cluster state (data not shown). 4.2 Physiologically Realistic Synaptic Connectivity. In the local cortical circuitry, nearby pairs of FS interneurons are often connected either bidirectionally or unidirectionally by GABAergic synapses (Galarreta & Hestrin, 2002). We studied the FS-neuron network with a slightly more realistic synaptic connectivity than the previous all-to-all connectivity. We rewired the previous large-scale symmetric network such that 70% of FS neuron pairs have unidirectional GABAergic synapses and the rest have bidirectional ones. The directions of the unidirectional connections were determined randomly. Gap junctions are present between all the neuron pairs. (We refer to this network as the 70% asymmetric network.) The results obtained from numerical simulations are displayed in Figures 5b.1 and 5b.2, where the same initial phases as used in Figures 5a.1 and 5a.2 were employed, respectively. Despite the fact that the GABAergic connections are unidirectional, a sufficiently coherent initial state led to a near-synchronous final state, as displayed in Figure 5b.1. For the present values of the parameters (Iext = 1.7 µA/cm2 and gsyn /ggap = 10), a unidirectionally coupled twoneuron network should exhibit a finite phase difference in the stable state (see Figure 4b.2). There exists, however, no stable configuration that allows all neuron pairs to have the same small phase difference in an asymmetric large-scale network. As a result, near-synchronous firing was achieved in the steady state, as displayed in Figure 5b.1. If the coherence of the initial state was not sufficiently high, the largescale network evolved into a non-phase-locked state. Note that this state differs from the so-called randomly firing state, in which each neuron generates a Poisson spike train. In the non-phase-locked state, the individual neurons fire almost periodically, but relative spike times between neuron pairs distribute randomly and change slowly in time (see the spike counts at the bottom of the figure). Comparing Figure 5b.2 with Figure 5a.2, we find that unidirectional connections destabilize the two-cluster state generated in the case of symmetric connections. The instability of the twocluster state in the large-scale asymmetric network is analogous to the loss of bistability in an asymmetrically connected two-neuron network (compare Figure 4a.2 with Figure 4b.2). We note that a fully asymmetric network (all the GABAergic synapses are unidirectional) has qualitatively the same stable states as those of the 70% asymmetric network (data not shown). We demonstrate how the synchronous spiking behavior of the physiologically realistic 70% asymmetric network can be rapidly modulated by


2191

neuron #

100 80 60 40 20 0 0

50

100

Starting from asynchronous state

20000 20050 Injecting random current

1000 1050

10050 10100

Injecting synchronous current for 1 ms

25050 25100

35050 35100 35150

time [ms]

Figure 6: Synchronous firing can be rapidly induced in the 70% asymmetric network by a brief synchronous external input to FS interneurons, while an asynchronous input lasting 100 ms can resettle the network activity in the nonphase-locked state. (Here, the same values of the parameters as used in Figure 5 were employed.)

injecting either synchronous or random input current. In Figure 6, the network was initially in the non-phase-locked state. At 1000 ms, all the neurons simultaneously received a brief depolarizing current (Iext = 20 µA/cm2 ) for 1 ms (synchronous current). This input caused a rapid increase of the coherence of the network activity to achieve a near-synchronous state. Conversely, established synchronous firing could be desynchronized by an asynchronous input. In the simulation, a neuron was randomly selected every 1 ms and innervated by the previous brief depolarizing current (random current). The random input, which lasted for 100 ms, from 20,000 ms to 20,100 ms, resettled the network in the non-phase-locked state. This simulation demonstrates that the degree of synchrony in an FS interneuron

2192


network can change between the two levels, high and low, depending on the coherence level of excitatory inputs to FS interneurons. 5 Discussion We have investigated the combined effects of gap junctions and GABAergic synapses on the synchronous firing in networks of FS interneurons. We have incorporated the Kv3.1/3.2 K+ channels into our neuron model to realize high-frequency nonadapting firing of the FS interneurons. Applying the phase-reduction method to two electrically coupled FS interneurons, we have shown that the neuron pair exhibits synchronous firing as a unique frequency-independent stable state, unlike a similar pair modeled with Kv1.3 K+ channels (Sherman & Rinzel, 1992; Suzuki & Aoyagi, 2002; see also Chow & Kopell, 2000). The effects of GABAergic synapses on synchronization depend on the decay time constant of the inhibitory postsynaptic current. In our simulations, the decay time constant was fixed at 5 ms; in reality, it might be even smaller (Bartos et al., 2002; Galarreta & Hestrin, 2002). In this case, incorporation of sufficiently strong bidirectional GABAergic synapses makes antisynchronous firing of the FS-neuron pair stable. For physiologically plausible values of the chemical-to-electrical synaptic conductance ratio, synchrony and antisynchrony coexist in almost the entire frequency range of the gamma oscillation (see Figures 4a.2–4a.3). Cooperative effects of gap junctions and GABAergic synapses on the emergence of synchronous firing were also studied in two coupled leaky integrate-andfire neurons (Lewis & Rinzel, 2003). On the other hand, two unidirectionally connected FS interneurons do not show the bistability. If a unidirectional GABAergic synapse is introduced into an electrically coupled FS interneuron pair, they only display a phase-locked state with a small phase difference. Interestingly, the two neurons discharge near synchronously at the gamma frequencies even at unrealistically large values of gsyn /ggap (see Figures 4b.2–4b.3). At very high frequencies (100–200 Hz), however, the near-synchronous state disappears, and a non-phase-locked state is achieved for large values of the conductance ratio (data not shown). Similar bistability has been found in a large-scale network of FS interneurons, which are simultaneously connected by electrical and chemical synapses. When the neurons are fully and symmetrically connected by GABAergic synapses, we find the coexistence of a synchronous state and a two-cluster state. The latter is a counterpart of the antisynchronous state appearing in the two-neuron system. A linear stability analysis of the two-cluster state was conducted to confirm the consistency between the theoretical results and the numerical results. On the other hand, when all the neuron pairs are only unidirectionally connected by the GABAergic synapses (fully asymmetric network), the network shows a bistable state consisting of near-synchronous firing and non-phase-locked firing.


2193

These patterns of the connectivity describe two extreme cases. It was reported in the cerebral cortex that 60% to 80% of FS interneuron pairs had chemical synapses, and among them, 25% to 60% of the pairs are connected bidirectionally (Galarreta & Hestrin, 1999, 2001, 2002; Gibson et al., 1999). Therefore, we have studied a fully connected large-scale network in which 30% of all neuron pairs are bidirectionally connected by GABAergic synapses and the other 70% of pairs are unidirectionally connected. As in the case of fully asymmetric chemical connections, the network displayed both synchronous firing and non-phase-locked firing. In many cases (but not always), the dynamical behavior of large-scale networks can be understood by analogy with the behavior of the twoneuron network. For instance, if a two-neuron network shows synchrony, the large-scale network with the same values of parameters other than the connectivity also shows synchrony. Similarly, when a two-neuron network shows antisynchrony, the corresponding large-scale network shows a twocluster state: Each cluster behaves as if it were a single FS interneuron (see Figure 5a.2). In some cases, however, the steady state of a large-scale network is not a naive extension of that of a two-neuron network. For instance, a unidirectionally connected two-neuron network always exhibits a small phase difference (see Figure 4b.2), but the corresponding asymmetrically connected large-scale network shows near-synchronous firing and a nonphase-locked state (see Figures 5b.1 and 5b.2). It is speculated that the nearsynchrony in the large-scale network may arise from averaging effects over the neural population. We have shown that the bistability achieved by electrical gap junctions and GABAergic synapses is a characteristic feature of the information processing by a network of FS interneurons. As demonstrated in Figure 6, an excitatory input can switch the state of the interneuron network between synchronous firing and non-phase-locked firing. Such bidirectional transitions may be engaged in stimulus- and context-dependent grouping of those neurons at the gamma frequencies (Buzsáki et al., 1994; Singer & Gray, 1995). Where does the modulatory excitatory input to FS interneurons originate from? A possible origin is an ensemble of pyramidal neurons projecting to the FS interneurons in the local cortical circuitry. A particular class of neocortical pyramidal neurons, i.e., chattering neurons (Kang & Kayano, 1994; Gray & McCormick, 1996; Steriade, Timofeev, Durmuller, & Grenier, 1998; Aoyagi, Kang, Terada, Kaneko, & Fukai, 2002), shows gamma-frequency rhythmic bursting. Thus, chattering neurons may exert a synchronous and powerful excitatory drive on the interneuron network at the gamma frequencies. Simultaneously, the feedback input from the FS interneurons may modulate the coherence of pyramidal cell activity according to the status of the interneuron network. In fact, a computational model of chattering neuron networks has indicated that such networks in general entrain regular spiking pyramidal cells only very slowly in the gamma band (Aoyagi, Takekawa, & Fukai, 2003). We speculate that FS interneurons assist chat-

2194


tering neurons to entrain task-related local cortical circuits quickly in the gamma band, in which synchronization among chattering neurons is most coherent (the results will be reported elsewhere). Another source of the excitatory drive is the thalamocortical input. In fact, inhibitory interneurons including FS neurons are principal targets of the thalamocortical input and receive much more powerful thalamic excitation than excitatory cortical neurons (Gibson et al., 1999; Porter, Johnson, & Agmon, 2001). Since the thalamus is likely to play a central role in attentive processing (Crick, 1984; McAlonan, Brown, & Bowman, 2000; Sherman, 2001), and since the cortical activity of animals in an attentive state often exhibits coherent gamma oscillations (Engel, Fries, & Singer, 2001), FS interneurons seem to engage heavily in generating the highly coherent cortical state during attentive processing (Whittington et al., 1995; Traub et al., 2001; White, Banks, Pearce, & Kopell, 2000). Functional roles of the network of FS interneuron must be further explored by combined experimental and theoretical studies. Appendix A The FS neuron model we study discharges periodically in response to a sufficiently large external current. In general, when multiple dynamical systems, each exhibiting oscillations, are interconnected by weak couplings, the coupled system can be reduced to a system consisting only of the phase degrees of freedom using the phase reduction method (Kuramoto, 1984; Ermentrout & Kopell, 1984). Assuming that both the synaptic coupling and gap junction are weak, we apply this method to our model. Let us describe the dynamics of the symmetric network depicted in Figure 4a.1 in vector form, dX1 X1 ) + Igap (X1 , X2 ) + Isyn (X1 , X2 ), = F( dt dX2 X2 ) + Igap (X2 , X1 ) + Isyn (X2 , X1 ), = F( dt

(A.1) (A.2)

where Xi ≡ (Vi , mi , hi , ni , ri ), and Vi , mi , hi , ni , and ri represent the quantities V, m, h, n, and r defined in equations 2.1–2.6 and 2.9–2.11 for the ith neuron, F is the vector representing the right-hand sides of equations 2.1, 2.2, and 2.9, and Igap and Isyn are the interaction functions arising from the gap junction (described by equation 2.7) and the chemical coupling (described by equation 2.8), respectively. Applying the phase-reduction method, we obtain the following phase model, in which φ1 and φ2 are the phases of neuron 1 and neuron 2, respectively: dφ1 = ω + gap (φ1 − φ2 ) + syn (φ1 − φ2 ), dt

(A.3)


dφ2 = ω + gap (φ2 − φ1 ) + syn (φ2 − φ1 ), dt

2195

(A.4)

where gap(syn) (φi − φj ) =

1 T

0

T

dλZV (λ + φi )Igap(syn) (λ + φi , λ + φj ).

(A.5)

The interactions gap and syn are periodic functions of T, and ω is the natural frequency of the neurons. The quantity ZV is the voltage component of the and describes the response (phase shift) of the phase-response function Z neuron to small perturbations of the membrane potential. In one cycle of oscillation, ZV in general takes both positive and negative values. If a small excitatory input is applied to the phase of one neuron at a time when ZV is positive, then the next spike of that neuron will be advanced in time, whereas if it is applied at a time when ZV is negative, then the next spike will be retarded in time. ZV can be obtained by computing the solution to the adjoint equation (Ermentrout, 1996). We note that gap and syn depend on only the relative phase of the two neurons. Now, subtracting equation A.4 from equation A.3, we obtain a differential equation that governs the time evolution of phase difference φ ≡ φ1 − φ2 : dφ − = SYM (φ) ≡ gap (φ) − gap (−φ) + syn (φ) − syn (−φ). dt

(A.6)

Similarly, for an asymmetric network (illustrated in Figure 4b.1), which is described by equations A.3 and A.4 with the syn term removed from the latter, the following differential equation is obtained: dφ − (φ) ≡ gap (φ) − gap (−φ) + syn (φ). = ASYM dt

(A.7)

If two weakly coupled neurons are phase-locked with a phase difference φ, the equation − (A)SYM (φ) = 0

(A.8)

must be satisfied. Although there may be several solutions to equation A.8, only those solutions that also satisfy the stability condition − d (A)SYM

dφ

(φ) < 0,

(A.9)

can appear in the equilibrium state. It is noted that gap (φ) and syn (φ) are proportional to ggap and gsyn , respectively, and therefore the conductance ratio gsyn /ggap is an effective parameter that affects the phase difference

2196


in the equilibrium state defined by equation A.8. In addition to the conductance ratio, the amplitude of the external current, Iext , is an important parameter that also affects the phase difference. Because Iext influences the phase-response function ZV through the modulation of the periodic solution of the membrane potential in an oscillatory cycle, ZV in turn defines gap(syn) (φ) according to equation A.5. Acknowledgments We thank T. Kaneko for helpful discussions. This work was partially supported by Japanese Grant-in-Aid for Science Research Fund from the Ministry of Education, Science and Culture (No. 13015024). References Aoyagi, T., Kang, Y., Terada, N., Kaneko, T., & Fukai, T. (2002). The role of Ca2+ dependent cationic current in generating gamma frequency rhythmic bursts: Modeling study. Neuroscience, 115, 1127–1138. Aoyagi, T., Takekawa, T., & Fukai, T. (2003). Gamma rhythmic bursts: Coherence control in networks of cortical pyramidal neurons. Neural Comput., 15, 1035– 1061. Bartos, M., Vida, I., Frotscher, M., Meyer, A., Monyer, H., Geiger, J., & Jonas, P. (2002). Fast synaptic inhibition promotes synchronized gamma oscillations in hippocampal interneuron networks. Proc. Natl. Acad. Sci. USA, 99, 13222– 13227. Beierlein, M., Gibson, J. R., & Connors, B. W. (2000). A network of electrically coupled interneurons drives synchronized inhibition in neocortex. Nat. Neurosci., 3, 904–910. Buzsáki, G., Llinás, R., Singer, W., Berthoz, A., & Christen, Y. (1994). Temporal coding in the brain. Berlin: Springer-Verlag. Chow, C. C., White, J. A., Ritt, J., & Kopell, N. (1998). Frequency control in synchronized networks of inhibitory neurons. J. Comput. Neurosci., 5, 407– 420. Chow, C. C., & Kopell, N. (2000). Dynamics of spiking neurons with electrical coupling. Neural Comput., 12, 1643–1678. Cobb, S. R., Huhl, E. H., Halasy, K., Paulsen, O., & Somogyi, P. (1995). Synchronization of neuronal activity in hippocampus by individual GABAergic interneurons. Nature, 378, 75–78. Connors, B. W., & Gutnick, M. J. (1990). Intrinsic firing patterns of diverse neocortical neurons. Trends Neurosci., 13, 99–104. Crick, F. (1984). Function of the thalamic reticular complex: The searchlight hypothesis. Proc. Natl. Acad. Sci. USA, 81, 4586–4590. Destexhe, A., Mainen, Z. F., & Sejnowski, T. J. (1994). An efficient method for computing synaptic conductances based on a kinetic model of receptor binding. Neural Comput., 6, 14–18.


2197

Durstewitz, D., Seamans, J. K., & Sejnowski, T. J. (2000). Dopamine-mediated stabilization of delay-period activity in a network model of prefrontal cortex. J. Neurophysiol., 83, 1733–1750. Engel, A. K., Fries, P., & Singer, W. (2001). Dynamic predictions: Oscillations and synchrony in top-down processing. Nat. Rev. Neurosci., 2, 704–716. Erisir, A., Lau, D., Rudy, B., & Leonard, C. S. (1999). Function of specific K+ channels in sustained high-frequency firing of fast-spiking neocortical interneurons. J. Neurophysiol., 82, 2476–2489. Ermentrout, G. B. (1996). Type I membranes, phase resetting curves, and synchrony. Neural Comput., 8, 979–1001. Ermentrout, G. B., & Kopell, N. (1984). Frequency plateaus in a chain of weakly coupled oscillators. SIAM J. Math. Anal., 15, 215–237. Galarreta, M., & Hestrin, S. (1999). A network of fast-spiking cells in the neocortex connected by electrical synapses. Nature, 402, 72–75. Galarreta, M., & Hestrin, S. (2001). Electrical synapses between GABA-releasing interneurons. Nat. Rev. Neurosci., 2, 425–433. Galarreta, M., & Hestrin, S. (2002). Electrical and chemical synapses among parvalbumin fast-spiking GABAergic interneurons in adult mouse neocortex. Proc. Natl. Acad. Sci. USA, 99, 12438–12443. Gerstner, W., van Hemmen, J. L., & Cowan, J. D. (1996). What matters in neuronal locking? Neural Comput., 8, 1653–1676. Gibson, J. R., Beierlein, M., & Connors, B. W. (1999). Two networks of electrically coupled inhibitory neurons in neocortex. Nature, 402, 75–79. Golomb, D., Hansel, D., Shraiman, B., & Sompolinsky, H. (1992). Clustering in globally coupled phase oscillators. Phys. Rev. A, 45, 3516–3530. Gray, C. M., & McCormick, D. A. (1996). Chattering cells: Superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex. Science, 274, 109–113. Hansel, D., Mato, G., & Meunier, C. (1993). Clustering and slow switching in globally coupled phase oscillators. Phys. Rev. E, 48, 3470–3477. Hansel, D., Mato, G., & Meunier, C. (1995). Synchrony in excitatory neural networks. Neural Comput., 7, 307–337. Kaneko, K. (1990). Clustering, coding, switching, hierarchical ordering and control in a network of chaotic elements. Physica D, 41, 136–172. Kanemasa, T., Gan, L., Perney, T. M., Wang, L.-Y., & Kaczmarek, L. K. (1995). Electrophysiological and pharmacological characterization of a mammalian Shaw channel expressed in NIH 3T3 fibroblasts. J. Neurophysiol., 74, 207–217. Kang, Y., & Kayano, F. (1994). Electrophysiological and morphological characteristics of layer VI pyramidal cells in the motor cortex. J. Neurophysiol., 72, 578–559. Kori, H., & Kuramoto, Y. (2001). Slow switching in globally coupled oscillators: Robustness and occurrence through delayed coupling. Phys. Rev. E, 63, 046214. Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. New York: Springer-Verlag. Lewis, T. J., & Rinzel, J. (2003). Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J. Comput. Neurosci., 14, 283–309.

2198


Lytton, W. W., & Sejnowski, T. J. (1991). Simulations of cortical pyramidal neurons synchronized by inhibitory interneurons. J. Neurophysiol., 66, 1059– 1079. McAlonan, K., Brown, V. J., & Bowman, E. M. (2000). Thalamic reticular nucleus activation reflects attentional gating during classical conditioning. J. Neurosci., 20, 8897–8901. Okuda, K. (1993). Variety and generality of clustering in globally coupled oscillators. Physica D, 63, 424–436. Porter, J. T., Johnson, C. K., & Agmon, A. (2001). Diverse types of interneurons generate thalamus-evoked feedforward inhibition in the mouse barrel cortex. J. Neurosci., 21, 2699–2710. Rinzel, J., & Ermentrout, G. B. (1989). Analysis of neural excitability and oscillations. In C. Koch & I. Segev (Eds.), Methods in neuronal modeling: From synapses to networks. Cambridge, MA: MIT Press. Rudy, B., & McBain, C. J. (2001). Kv3 channels: Voltage-gated K+ channels designed for high-frequency repetitive firing. Trends Neurosci., 24, 517–526. Sherman, S. M. (2001). Thalamic relay functions. Prog. Brain Res., 134, 51–69. Sherman, A., & Rinzel, J. (1992). Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proc. Natl. Acad. Sci. USA, 89, 2471–2474. Singer, W., & Gray, C. M. (1995). Visual feature integration and the temporal correlation hypothesis. Annu. Rev. Neurosci., 18, 555–586. Steriade, M., Timofeev, I., Durmuller, N., & Grenier, F. (1998). Dynamic properties of corticothalamic neurons and local cortical interneurons generating fast rhythmic (30–40 Hz) spike bursts. J. Neurophysiol., 79, 483–490. Suzuki, D., & Aoyagi, T. (2002). Phase locking states in network of inhibitory neurons: A putative role of gap junction. J. Phys. Soc. Jpn., 71, 2644–2647. Traub, R. D., Kopell, N., Bibbig, A., Buhl, E. H., LeBeau, F. E. N., & Whittington, M. A. (2001). Gap junctions between interneuron dendrites can enhance synchrony of gamma oscillations in distributed networks. J. Neurosci., 21, 9478–9486. van Vreeswijk, C., Abbott, L. F., & Ermentrout, G. B. (1994). When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci., 1, 313–321. White, J. A., Banks, M. I., Pearce, R. A., & Kopell, N. J. (2000). Networks of interneurons with fast and slow γ -aminobutyric acid type A (GABAA ) kinetics provide substrate for mixed gamma-theta rhythm. Proc. Natl. Acad. Sci. USA, 97, 8128–8133. White, J. A., Chow, C. C., Ritt, J., Soto-Trevino, ˜ C., & Kopell, N. (1998). Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. J. Comput. Neurosci., 5, 5–16. Whittington, M. A., Traub, R. D., & Jefferys, J. G. R. (1995). Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature, 373, 612–615. Received June 25, 2002; accepted February 28, 2003.