Figure 1.
Distinct oscillation frequencies appearing intermittently in time, and schematic representation of model networks.
a. Wavelet of EEG recordings in the frontal zone of the human brain, showing two interspersed, non-harmonic frequencies (17.7±0.8 Hz and 22.9±0.8 Hz; white arrows) in the beta range. Color indicates power of oscillations. When one of the frequencies has high power, the other oscillation frequency is absent or has low power. Adapted from [26] (Fig. 12 therein). b. Intracranial field recordings in the medial prefrontal cortex of awake rats show similar dynamics. The two main frequencies are 15.8±0.3 Hz and 22±1.7 Hz (white arrows). Adapted from [26] (Fig. 12 therein). c. Schematic representation of the two model networks, each consisting of a population of excitatory cells (e, E) and a population of inhibitory cells (i, I). The network generating slow oscillations is labelled with lower case letters, and the network producing fast oscillations is labelled with upper case letters. In each network, the inhibitory cells projected among each other and to the excitatory cells. Likewise, the excitatory cells projected among each other and to the inhibitory cells. In addition, both cell types received external input in the form of a constant depolarizing current (CDC). Furthermore, the cells of one network projected to the cells of the other network (not shown). The different inter-network connectivity schemes studied are shown in Fig. 2.
Figure 2.
The connectivity schemes between the two model networks.
In the slow network, the excitatory and inhibitory populations are labelled with lower case letters (e, i) and in the fast network with upper case letters (E, I). Each column (a–d, A–D) comprises what we call a connectivity class, consisting of eight different connectivity schemes. The strength of the connectivity type shown in red was varied in the simulations. A connectivity class is labelled with a lower or upper case letter depending on whether the slow or the fast network, respectively, is the network projecting to the other network. (See further Methods.)
Figure 3.
Oscillatory activity in the slow and the fast network when they are unconnected or connected.
Shown are the Fourier transform (a, d, g), wavelet transform (b, e, h) and raster diagram of cell firing (c, f, i) of the excitatory population in either the slow or the fast network. Color in the wavelet transforms indicates power of oscillation. The raster diagrams depict the firing times (indicated by dots). a–c. Activity of the slow network in isolation. The Fourier transform shows peaks at the base frequency (20.4 Hz) and at the first and second harmonics. Owing to the highly synchronized activity (making the signal effectively a comb function), the Fourier transform produced peaks at the harmonics, but there were no cells that actually fired at these frequencies (see panel c). d–f. Activity of the fast network in isolation. The Fourier transform shows peaks at the base frequency (32.4 Hz) and the first harmonic. g–i. Activity of the fast network when the excitatory cells of the slow network projected to the excitatory cells of the fast network (eE connection) with conductance factor (see Methods). With this connection strength, the slow network managed to impose its rhythm onto the fast network, in which the base frequency (20.4 Hz) of the slow network and its first harmonic were strongly expressed. Since there were no connections from the fast to the slow network, the activity of the slow network was not different from that in the unconnected situation.
Figure 4.
The eE connections from the slow to the fast network can impose the slow rhythm onto the fast network already at low connection strengths.
Lower case letters (e, i) label the excitatory and inhibitory populations in the slow network, and upper case letters those in the fast network (E, I). The strength (conductance factor) of the eE connections (red arrow in connection scheme) is varied in the different connectivity schemes. The blue discs indicate the oscillation frequencies in the fast network; their diameters depict the power. The red dots show the base frequency and the first harmonic of the slow network, without indicating power. The green arrow points to the base frequency of the slow network. In all connectivity schemes, for high eE connection strengths, the fast network became frequency locked to the rhythm of the slow network, at its base frequency and/or at the corresponding first harmonic. For lower connection strengths, the base frequency of the slow network and a frequency close to the base frequency of the fast network coexisted in the fast network (e.g., a7, a8).
Figure 5.
The eI connections from the slow to the fast network can impose the slow rhythm onto the fast network already at low connection strengths.
In all connectivity schemes, the fast network oscillated at the base frequency of the slow network (and its first harmonic) for high eI connection strength. For lower strengths, two different frequencies could coexist (b4, b6, b8).
Figure 6.
The effect of inter-network connections on firing pattern.
Shown are, for different connectivity schemes, the Fourier transform and the raster diagram of cell firing in the excitatory population of the target network. The conductance factor Cf indicates the relative strength of the red connection (see Methods). Panels a–f show connectivity schemes a2, A2, c2, B2, a8 and B8, respectively (see Fig. 2). In a–d, the source network completely imposed its rhythm onto the target network, whereas in e and f two different non-harmonic frequencies could coexist. In c and d, increased inhibition in the target network due to strong iE or Ei connections reduced the power of oscillatory activity in the target network.
Figure 7.
The Ee connections from the fast to the slow network can impose the fast rhythm onto the slow network, especially at high connection strengths.
The blue discs now indicate the oscillation frequencies in the slow network; their diameter depicts the power. The red dots show the base frequency and the first harmonic of the oscillatory activity in the fast network, without indicating power. The green arrow points to the base frequency of the fast network. For high Ee connection strength, the slow network became fully entrained to the rhythm of the fast network. For intermediate Ee connection strengths, some connectivity schemes (e.g., A6) exhibited an activity pattern in which two different oscillation frequencies coexisted.
Figure 8.
Continual expression of coexistent oscillation frequencies.
Shown are the Fourier transform (a), wavelet transform (b) and raster diagram of cell firing (c) from the excitatory population of the target network (the fast network). The inset in a shows the connectivity scheme from the slow to the fast network (see Fig. S1c8), in which the iE connection had . The fast network co-expressed its own fast base frequency (32.4 Hz) and the base frequency of the slow network (20.4 Hz and corresponding harmonic and subharmonic frequencies). The power (amplitude) of both frequencies did not vary strongly over time.
Figure 9.
Alternating expression of coexistent oscillation frequencies.
Shown are the Fourier transform (a, d), wavelet transform (b, e) and raster diagram of cell firing (c, f) from the excitatory population of the target network (the fast network). The insets in a and d show the connectivity schemes from the slow to the fast network (see Figs. 4a5 and S1c4, respectively), in which the eE connection (a) had and the iE connection (d) had
. For both connectivity schemes, the base frequency of the fast network (32.4 Hz) and the base frequency of the slow network (20.4 Hz) appeared intermittently in the fast network. In b, the frequency switched faster than in e. When either frequency component was present, its power remained relatively constant.
Figure 10.
Expression of a single oscillation frequency with strong fluctuations in power.
Shown are the Fourier transform (a, d), wavelet transform (b, e) and raster diagram of cell firing (c, f) from the excitatory population of the target network (the slow network in a–c and the fast network in d–f). The inset in a shows the connectivity scheme from the fast to the slow network (see Fig. S4C2), in which the Ie connection had . In the time interval shown, the slow network expressed its own base frequency (20.4 Hz) but with strong fluctuations in power. The inset in d shows the connectivity scheme from the slow to the fast network (see Fig. 5b2), in which the eI connection had
. The fast network expressed the base frequency of the slow network with strong fluctuations in power.
Figure 11.
Alternating expression of coexistent oscillation frequencies with fluctuations in power.
Shown are the Fourier transform (a), wavelet transform (b) and raster diagram of cell firing (c) from the excitatory population of the target network (the fast network). The inset in a shows the connectivity scheme from the slow to the fast network (see Fig. S1c5), in which the iE connection had . The base frequency of the fast network (32.4 Hz) and the base frequency of the slow network (20.4 Hz) appeared more or less intermittently in the fast network. When either frequency component was present, its power was not stable over time (e.g., the slow base frequency between t = 33.7 s and t = 33.9 s).
Figure 12.
Alternating expression of two coexistent oscillation frequencies in rat PFC.
Wavelet transform of extracellular field potential in layer 5 of rat PFC slice. Two different non-harmonic frequencies, around 7 Hz and 13 Hz, alternate as dominant frequency but are occasionally expressed simultaneously (e.g., around t = 5.5 s). The figure was not published before but is based on data collected in [26].